Youtube daily Jan 13 2018

Welcome to the first Mathologer video of the year. Today it's about something very

serious and so I'm wearing a totally black t-shirt. You all like Numberphile

right? Me too, except for this one video over there in which they prove the

infamous identity 1+2+3+...=-1/12

using some simple algebra that even kids in primary school should be

able to follow. Since this video was published in 2014 over six million

people have watched it and more than 65,000 have liked it. Unfortunately,

pretty much every single statement made in this video is wrong. And by wrong I

mean wrong in capital letters. In particular, as anybody who knows any

mathematics will confirm 1+2+3+... sums to exactly what common sense

suggests it should namely plus infinity. And this video was not published on the

1st of April. Also, as we all know, the Numberphile videos are presented by

smart guys, in this case university physics professors, who do know their

maths and who are definitely not out to mislead us. So how did they get it so

horribly wrong and what did they really want to say. Well, they started out with

some genuinely deep an amazing connection between 1 + 2 + 3 etc and the

number -1/12 but in the effort to explain this connection in really, really

simple terms they just went overboard and ended up with an explanation that is

not just really simple but also really wrong. Well 6 million views later and the

comment sections of all maths YouTubers are being inundated by confused one plus

two plus three comments that are a direct consequence of this video. For

mathematical public relations it's a disaster. (Marty) It's THE disaster. (Mathologer) Yeah, it's THE disaster. And so I think

it's a good idea to have another really close look at the Numberphile

calculation step by step, state clearly what's wrong with it, how to fix it, and

how to reconnect it to the genuine maths that the Numberphile professors had in

mind originally. Lots of amazing maths look forward: non-standard

summation methods for divergent series, the eta function, a very well-behaved

sister of the Zeta function, the gist of analytic continuation in simple words,

some more of Euler's mathemagical tricks etc. Now, I've tried to make this whole thing

self-contained. So you don't have to have watched my very different other video on

one plus two plus three from over a year ago or anything else to understand this

one. Okay, let's get going. So that we are all on the same page, here real quick is

the whole Numberphile calculation. They call the unknown value of the

infinite series 1 + 2 + 3+... S. As stepping-stones for the calculation they

first calculate the sums of these other two infinite series. So 1-1+1-1+...

and 1-2+3-4+... Adding up the terms of

the first series, we get the partial sums 1. Ok 1 minus 1 is 0, 1 minus 1 plus 1 is

1, 1 minus 1 plus 1 minus 1 is 0 and so on. These partial sums alternate between

0 and 1 and so the Numberphile guys declare that the sum of this infinite

series is CLEARLY the average of 1 and 0 which is 1/2 (Marty) That's not all that clear to me. (Mathologer) Alright we'll get

to that. They also mention that there are other ways to justify this. We'll

also get to that. Ok, second sum. Here they start by considering what happens when

you double this sum. So 2 times S2 is equal to the infinite series added to

itself but now before adding the two infinite series they shift the bottom

series one term to the right. Now 1 plus nothing is 1, minus 2 plus 1 is minus 1, 3

minus 2 is 1, minus 4 plus 3 is minus 1, etc. But that bottom series is the one we

already looked at which, remember, is equal to 1/2 and so ....

Second sum done, great. Now for last sum, that's the one we're really after. Here the Numberphile

guys start by subtracting S2, the sum they just figured out, from S. Now 1

minus 1 is 0, 2 minus minus 2 is plus 4, 3 minus 3 is 0, 4 minus minus 4 is 8, etc.

The zeros don't matter so let's get rid of them. Take out the common factor 4. Ah

the yellow that's our 1+2+3+... sum S again. Now solve for S, and my

usual magic here, and we get -1/12. And here the Numberphile guys take a bow.

But, not so fast! !ll this is really nonsense the way it was presented. In

particular these three identities are false. This means that if on any maths

exam at any university on Earth you're asked to evaluate the sums of these

infinite series and you give the Numberphile identities as your answer

you will receive exactly 0 marks for your answers. It's critical to realise

that in mathematics we have a precise definition which underpins the sum of an

infinite series. Wherever you see infinite series this definition and only

this definition applies unless there are some huge disclaimers to the contrary in

flashing neon lights. Alright, now the Numberphile guys did not include any

such disclaimers and so they too should get 0 marks for their effort. (Marty) Or maybe give them -1/12 marks. (Mathologer) Yeah I

think I can agree with that. OK, what's this definition and what are the

answers that will get you full marks on your maths exam. To evaluate the sum of

an infinite series you calculate the sequence of partial sums

just like the Numberphile guys did at the very beginning. Now if the sequence

of partial sums levels off to a finite number, that is, if the sequence converges,

or if it explodes to plus infinity, or if it explodes to minus

infinity, then this limit is the sum of the infinite series. If no such limit

exists, then the infinite series does not have a sum. That's it, that's the

definition. So for the first Numberphile series the sequence of partial sums

alternates between 0 and 1 and therefore does not have a limit. This means that

this infinite series does not have a sum, neither 1/2 nor anything else. This is

the correct answer for your maths exam. Alright, what about the other two

infinite series? Hmm, well, in the case of 1 plus 2 plus 3 the partial sums

explode to plus infinity and so the sum of the series

is infinity. For the infinite series in the middle the partial sums explode in

size, but neither just to infinity or just to minus infinity, and so this

series also does not have a sum. So these are the answers that get you full marks.

In many ways the most important infinite series are those with a finite

sum which have not featured here yet. So, to give some perspective, here's a

standard example, an infinite geometric series: 1/2+1/4+1/8 and so on.

Now here the partial sums exhibit a really nice pattern and

clearly they converge to 1. (Marty) Yeah, I think this one is clear. (Mathologer) This one is clear and so the sum of this infinite

series is 1. Oh, before I forget, those finite sum series are usually called

convergent series and all the other infinite series are called

divergent series. Keeping this in mind let's have another look at the

Numberphile calculation. Here's the whole thing at a glance. It's just a transcript

of the writing on the brown paper in the Numberphile video. Again, as it was

presented by Numberphile all this is nonsense and worth 0 marks. (Marty) Or less! (Mathologer) Or less :) THIS.

IS. NOT. MATHEMATICS. Don't use it, otherwise you'll burn in mathematical

hell. Having said that there should be some

method to this madness, right? Those guys are smart! But if there is,

then it's clear that the sums you see here cannot possibly represent the usual

sums, as about six million people have been misled to believe by this video. Ok

let's start by doing something that may also seem a little bit crazy. At first

glance, just for fun, and in denial of reality, let's assume for a second that

those three Numberphile series were actually convergent, that is, all had a

finite sum. Then all, ALL highlighted arguments would be valid. This includes

the termwise adding and subtracting of series that was performed here, ... and here, ...

and even the shifting to the right before the addition that a lot of people

view with suspicion. Why would all these operations be ok if we were dealing with

convergent series? Because summation of convergent series is consistent with

termwise addition and subtraction, and shifting. Let me explain that too. There

are differences between finite and infinite sums. For example, infinite

series sometimes don't have a sum whereas finite sums always exist,

rearrangement of the terms can change the sum of convergent infinite series,

etc. On the other hand, the sums of convergent infinite series do share a

lot of the properties of finite sums and it's exactly these properties that make

them so useful. Here the most important three such properties. Let's say you have

two convergent infinite series, okay< with sums A and B. Then, by adding these

two series termwise you get a new infinite series. And now it's quite easy

to prove that this new infinite series is also convergent and that its sum

is equal to A plus B, of course. And the same stays true if you replace all the

pluses with minuses. So termwise addition and subtraction is consistent with

summing convergent series. That's property one. Property two. Multiplying the

terms of a convergent series with sum A by a number, say

five, gives a new infinite series. Again it's really easy to see that the new

series is convergent and that its sum is five times A. So termwise

multiplication by numbers is also consistent with summing convergent

infinite series. That's our second property. Finally, property three. Shifting

the terms of a convergent series with sum A by one term is the same as adding

a zero as the first term to our series, like that. Obviously, the new series is

still convergent and its sum is the same as that of the original series. This also

works the opposite way. Removing a zero term at the front does not change the

sum. Okay, so that's property three. Now we can use these three properties to build

valid arguments, very much like in the Numberphile video. Here's an example. This

is the convergent series that I showed you earlier.

Remember its sum is 1. Now let's say we did not know its sum or even whether

this series is convergent or not. Then we could argue in a legit way like this: Ok,

well we don't know whether it's convergent or not, but if it's convergent

and it's sum is M (M for mystery number :) then because of the number multiplication

property we get that half M is equal to 1/2 times 1/2 which equals to 1/4 plus

1/4 times 1/2 which is equal to 1/8 and so on. But because of the shift property

1/2 M is also equal to this guy here zero plus whatever. Now because of the

addition and subtraction property we are justified to subtract as in the Numberphile

video. So, on the left side we get M minus 1/2 M, that's 1/2 M, and on the

right side we've got 1/2 minus zero that's 1/2, and then everything else kind

of goes away. In total we get M is equal to 1. So our assumption that our

mystery series is convergent with sum M lets us conclude in a valid way

that the only value that M can possibly be is 1. Question: Does this prove that

M is 1? No, and this is very important. Because this argument starts with an

assumption, to be able to conclude that the sum of the series really is 1, you

still have to somehow show that the series we started with is actually

convergent. Hmm so this sort of argument gives you an idea of what to expect but

it does not get you all the way. Anyway let's see what happens when we unleash

this sort of reasoning on the first Numberphile series which we already know

is NOT convergent. Well, so just for kicks, let's assume that the 1-1+1

series was actually convergent with the unknown sum S1. Then, because of the

shift property, S1 would also be equal to 0 plus all the other junk. Again,

because of the addition property, we can add on both sides S1 plus S1 that's

2 S1 and then we add termwise on the right to get, well, 1 there and everything

else cancels out as you can see. And so we get this which is exactly what

Numberphile said S1 should be. So the plot thickens here, right? In fact, in

some ways this line of reasoning would have fitted in better with the rest of

their argument than just plucking the number 1/2 out of thin air. So let's

have another look at the Numberphile argument and let's fit in what we just

did as the first step to justify why S1 should be 1/2. So here we go. This is the

Numberphile argument. Let's round it off by inserting the argument for S1 from

just a moment ago. Here we go. And so here is one way to rephrase the

whole thing to make it into a valid argument. IF, and this is a big, huge

monstrous IF, these three infinite series were convergent, THEN this whole argument

would be valid and the sums of these three infinite series would be exactly the

numbers given by Numberphile. Great, but of course we know that the assumption of

this valid argument is false, that the three infinite series are not convergent.

So, yes, this argument is valid in itself, but what

good is it if the assumption is starts with is false? Well here's an idea.

Since no divergent series has a finite value attached to it, let's dream big.

What if it was possible to extend the notion of summing a

convergent series, what if it was possible to define a super sum. This

super sum should have three key properties: first of all, we don't want to lose

anything so the super sum of a convergent series should be the same as

its normal sum, right? Then all divergent infinite series should be super summable

with finite values. And then the last thing we want is that super summing, just

like normal summing is consistent with adding, shifting, etc. alright. Now such a

super sum extension of the standard infinite sum would be as fantastic as

the extension of the real numbers to the complex numbers, with all sorts of cool

properties for a smaller world remaining true in the bigger world and at the same

time all sorts of new magic appearing in the bigger world. In particular, because

consistency made the argument over there valid for convergent series, we would

expect the argument to still be valid if we replaced ordinary equality by equality

with respect to super summing. Even better, since every infinite series would

have a super sum, we could get rid of the if and so the whole Numberphile video

could be saved by just saying that we are super summing instead of just boring

old summing. What a lovely dream :) (Marty) It's time to wake up! (Mathologer) Yes, sadly. Well, anyway, those of you who watched my

last video on this topic know that there is a super sum. However it only assigns

a sum to some :) divergent series but not to all. In particular, it sums the first

two guys over there. To show how super summing works let's apply it to our first

annoying divergent series. Basically super summing builds upon normal summing

and then averaging out any bouncing around of the

partial sums. We start by calculating the sequence of partial sums. If this

sequence converges, then our super sum equals our normal sum and no tricks are

needed. However if the sequence of partial sums does not converge, as in the

case of this infinite series, then we start with the trickery, building a second

sequence out of the first. The Nth term of this new sequence is just the average

of the first N terms of the first sequence. So, for our particular series

for the first term of the new sequence there's nothing to average, well the

average of 1 is just 1. Ok, then the average of 1 is 0 is 1/2, the average of

1 and 0 and 1 is 2/3, etc. Now, every second number here is 1/2 and the remaining

numbers also converge to 1/2 and this means that overall the second series

converges to 1/2. And that means that our infinite series super sums to 1/2, which

is also the number that Numberphile gets. Now, for other infinite series even the

second sequence may not converge in which case we generate a third sequence,

again by averaging the second sequence. If that doesn't work, then we

generate a fourth, then a fifth, etc. As long as any of those sequences

converges, the infinite series under discussion has the corresponding limit

as super sum. For example, in the case of the second Numberphile series, the first

and second sequences diverge but the third sequence converges to 1/4

which then is our super sum. This is also what Numberphile gets and so everything

is looking good. Until now, now we have to confront the sad truth that for most

infinite series none of the associative averaging sequences of numbers converge

and so these series don't have a super sum. In particular, for 1 plus 2

plus 3 and so on all the partial sums are positive and obviously averaging

over positive numbers will always only result in positive numbers.

In fact, all the associated sequences of numbers will explore to positive

infinity and so 1 + 2 + 3 etc definitely does not super sum to anything finite,

let alone anything finite and negative like -1/12. Returning to the

Numberphile calculation, here's the part that can be totally justified using

super sums instead of regular sums. Because not every divergent series has a

super sum we still need the big IF to make this part of the argument valid. In

itself not too bad, though, since the assumption is actually true, right? And

since super summing is really the most natural extension of normal summing, 1/2

and 1/4 are the only reasonable numbers to associate with the first two series.

Really nice stuff I think. To recap, we now know finite sums, convergen

infinite sums and our new super sums. Oh, by the way, I should mention that in the

literature our super sum would be called generalized Cesaro summation or

generalized Hölder summation. Anyway, these summation methods are proper

extensions of each other and are therefore able to assign meaningful

values to larger and larger classes of series. However, being able to do more

also comes at a price. The more powerful a summation method, the less well behaved

it is. What works for finite sums cannot necessarily be taken for granted for

infinite sums. I already mentioned problems with rearranging convergent series, for

example. Of course, super sums also lack all the nice summy properties that

normal infinite sums lack but they are even summy things that still work for

infinite sums that no longer work for our super sums.

Yes, yes the three basic properties I've been hammering are fine but to assume

that any familiar summy property will also work for super sums in general is

risking zero marks (Marty): Or less! (Mathologer) Or less :) For example, inserting or deleting

infinitely zeros has no effect on convergent series. However, doing the

same to super sums can change things dramatically. For example, if we insert

infinitely many zeros into our first series, like this, the super sum of the

new series will be different from 1. Little puzzle for you:

What's the new super sum? As usual, give your answers in the comments. And this

zero problem is important. I glossed over this because it won't have any bearing

on our discussion, but at some point in the Numberphile calculation they simply

zap infinitely many zeros and this cannot be justified with our three

properties. Bad... The effect of losing more and more properties as you go more and

more general is actually something that you've all encountered before when you

got introduced to larger and larger number systems: fFrst the positive

numbers, then to the integers, the rational numbers, the reals and to the

complex numbers and even beyond to the quarternions and the octonions. Each

time you broaden your world, you lose some nice properties. Second puzzle for

you: Can you think of some properties that get lost along the way as we build

larger and larger number systems? And another puzzle: Suppose we assume that

the 1 + 2 + 3 etc series actually super sums to a finite number, can you

manipulate this identity into a couple of contradictory statements using our

favourite three properties? What can you then conclude from the fact

that you can arrive at contradictory statements? It will be interesting to see

what you can come up with in this respect? Anyway, it's time! (drum roll)

We have to get serious about the connection between our 1 + 2 + 3 series

and - 1/12. So press the pause button, go get your popcorn and your hot

chocolate and let me know when you're back :) (jeopardy music) Ready? Here we go.

Even at the level of super sums we are pretty far removed from what most people

think of as a sum. After all for divergent series, given all the averaging

that is going on, the super summing is really more like finding a super average

than a real summy sum, don't you think? Well it will get more extreme not only in this

respect but also in terms of the maths that is required to understand what is

going on with the -1/12 connection. I'm sure that a lot of you

will already have heard of this connection, so let me just state it first

and then really explain it using the Numberphile calculation as a template.

This is the mega famous Riemann zeta function. It is a function of the complex

variable z. Written as the infinite sum there it makes sense if the real part of

z is greater than 1. However, there is one unique way to extend the Riemann zeta

function to an analytic function for all complex numbers z excepting 1.

Formally, if you substitute z = -1 you get ... Well, of course, minus 1

is not greater than 1, and so we really don't have equality here, and so let's

quickly get rid of their equal sign. Ok, at the same time the right-hand side is

our master villain 1 + 2 + 3 etc and the value of zeta

at minus 1 is equal to you guessed it - 1/12. And this, in a nutshell, is the

genuine, real, actual connection between 1 + 2 + 3 + and - 1/12. But why would

anybody describe this connection as a sum, and what has all this to do wit the

last part of the Numberphile calculation. Well there's more to explain. First,

here's a mini introduction to analytic functions and analytic continuation. This

will be a rough and ready intro which is all we need. You all know what a

polynomial is, right? One of these guys: a constant function or a linear function

or a quadratic function or a cubic, etc. Now let's play a game. Here is a chunk of

a mystery continuous function that is defined for all real numbers. So I'm just

not showing you the part to the left of the y-axis. Here's the question. Just by

looking at this chunk, can you continue the graph and tell me what my function

is? Now you might be tempted to say 'Yes' but the answer is 'No'. There are infinitely

many ways we could continue to the left and here are a couple of examples. Here's one and

there's another one, there's a third one, there's infinitely many different ones.

Were you tricked? (Marty) No. (Mathologer) Sure you were not , but you know the game, right? Of course our

initial chunk is part of a line and it's natural to think of continuing the

function as this same line. But we don't have to. However, if I tell you the

mystery function is linear, then your initial chunk tells you exactly how to

continue the function, there's only one way to continue so that the whole

function is linear. In fact, the same is true if I only told you the function was

a polynomial. Just by looking at the chunk you could be absolutely sure that

my polynomial is linear and exactly which linear function, right? We can now

generalize this simple observation a couple of steps, in a pretty dramatic way

actually. Here we go. First, suppose that our initial chunk is

part of a parabola, or if you like a cubic, or any polynomial. If I then tell

you that my mystery function is a polynomial there's always only going to

be exactly one polynomial that continues our beginning chunk. In other words, a

polynomial is completely determined by any part of it. Second, all we've said

stays true if we think of polynomials as functions of a complex variable and if

you begin with a chunk of the polynomial corresponding to a region in the complex

plane. So on the left, you see the complex number plane where each point stands for

a complex number and I've also colored a small region in

the plane. And so in terms of this picture a polynomial is completely

pinned down by the values it takes on over a region like this. No other

polynomial will take on all the same values there. Again, just relax if all this

seems a little bit too much. Now, the polynomials are the simplest and

most nicely behaved functions but there is a much larger world of functions that

shares a lot of the nicest properties with polynomials. Those are the so-called

analytic functions. These are the complex functions that can be expressed locally

as either regular finite polynomials or as infinite polynomials, so called power

series. For example, the exponential function is an analytic function because

it can be written as an infinite polynomial like this. In fact, pretty much

all our favorite functions such as the trig functions, rational functions, etc.

are analytic. Important for us is that just like a polynomial, an analytic

function is completely pinned down by any initial chunk. So if I give you a

beginning chunk of an analytic function like the exponential function, then no

other analytic function can continue this chunk. This is usually expressed by

saying that analytic continuation of an analytic function is uniquely determined.

In summary, though there may be many nice ad hoc ways to continue an analytic

function there's just one distinguished, most reasonable, absolutely fantastic

never to be improved way to do this, leading to a larger analytic function. Of

course, as I said, this is all very sketchy and you guys in the know will

probably nitpick me to death in the comments. (Marty) Looking forward to it. Ok, in particular I didn't tell

you why we need to drag complex functions into the discussion, but please

just run with it for today and I promise I'll fill in the details soon. In the

meantime you can also read up on things by following the links in the

description. All you really have to remember is this: an initial chunk of an

analytic function nails down the whole analytic function. Now we can join the

dots. We have two completely different notions of best extension. First, for

extending sums to super sums of divergent series

and second for extending a chunk of an analytic function to the whole analytic

function. Combining these two extension ideas, we can finally explain what's

going on with 1 + 2 + 3 + and -1/12. Okay, have a look at this infinite series.

Notice that it's the same as the zeta function except that it includes minuses.

It's also obviously different from the Numberphile series in that it includes a

variable z. So it is actually an infinite family of infinite series, one for each

complex number z. Let's just make a little list of such series corresponding

to a few prominent integer values. For z = 0 we

our 1 minus 1 plus 1 series. Ok for x = - 1 we get 1 minus 2 plus 3

and Mathologer regulars know that for z = 1 and 2 the series are

convergent. Now, in general, these series are convergent for all complex numbers z

in the positive brownish half plane. The infinite series are divergent for all

other z including 1 minus 1 plus 1 etc at 0 and the other guy. However, just like

the two Numberphile series can be super summed, the same is possible for every

z, for all the divergent series in this family. And this allows us to define a

close relative of the zeta function, the so-called

Dirichlet eta function. And this function turns out to be an analytic

function. So to start with, standard summation only gives us part of this

analytic function for which two infinite series converge, this part here. Now, most

mathematicians will simply discount a divergent series that pop up here as

useless artifacts. Instead they will construct the analytic continuation of the

eta function by completely different methods. These methods are very slick and

ingenious. However, they provide very little

intuition and insight into what's really going on here. On the other hand, seeing

that the most reasonable extension of an analytic

function that is defined on part of the complex plane is actually given by the

most reasonable way to assign generalised sums to these supposedly useless

divergent series just feels right to me and leads the way to a more intuitive

understanding of analytic continuation, at least in this case. But now here's a

great thought: we just used a generalised sum to construct an analytic

continuation, right? Let's turn the idea around, let's use analytic continuation

to identify candidates for a generalised summation method. And this is exactly

what happens in the case of 1 + 2 + 3 etc and - 1/12 and the zeta function. You

get the zeta function when you replace all the minuses in the eta function by

pluses. Well you get part of it, the brownish part. which is the part of the

complex plane for which the infinite series on the right converge. For all

other z the resulting infinite series are divergent and even super summing

doesn't help for the white part on the left. So the super summing trick for

eta just doesn't work for zeta. The trick to use is encoded in the finale of

the Numberphile pseudo proof. That's the brownish bit down there. Remember this

part of the argument takes the sum S2 of the 1 - 2 + 3 series and spits out a

sum for the 1 + 2 + 3 series formally these two sums are just what you get

when you let z equal to -1 in the infinite series of the eta and zeta

functions and actually the Numberphile calculation is just a special instance

of a calculation involving eta and zeta. Let's be brave. Ignoring questions

of legality, let's unleash exactly the same calculation on eta and zeta. So

instead of subtracting S2 from S, let's subtract eta

from zeta. Ready? ... Right, take out the common factor down there. That's zeta again in

the brackets. Now let's solve for zeta. There my magic

again okay that's really exactly the same as the last part of the Numberphile

calculation using zetas and eta instead of the Numberphile series. Just as a

check after substituting z = -1 this identity turns into

this, and with S2 being 1/4 we get this. Okay more magic and we're back to

-1/12. But didn't we say that the Numberphile computation was nonsense?

(Marty) Yes, we did. (Mathologer) We definitely did. And it is, but some magic happens with zetas and eta which

saves our zetas eta calculation from being nonsense and that is the magic of

analytic continuation. Both the series for zetas and eta are convergent for

every z in this brownish region. This means that for these values of z our

calculation and the resulting equation above are pure, correct, 100% approved

bona fide mathematics. But, as well, eta is defined and analytic for all z

and the same is true for the denominator. But then the right side, as a quotient

of two functions that are analytic everywhere, is itself defined and

analytic everywhere except possibly at the zeroes of the denominator. In fact, a

closer look reveals that the whole right-hand side is analytic everywhere

except at z =0. Here comes the punchline and this punchline

hinges on the chunks-pin-down-analytic-functions business that I've been going

on about. You should really try to understand this. Okay, so both the left

side and the right side are analytic in this part of the complex plane here. But

since the right side is analytic everywhere, because of our chunks-pin-

down-analytic-functions property, the right side has to be equal to the

elusive analytic continuation of the zeta function on the left that everybody is

really interested, the analytic continuation of the zetas function. So

this identity is a real jewel as it gives an explicit way to calculate any

value of the Riemann zeta function, analytic continuation and all via the eta

function which, remember, we defined via super summing. In other words, it actually

makes sense to use this identity as a definition for the zeta function which

works for all z. For example, setting z=0 we get this one here

and eta of zero was just the super sum of 1 minus 1 plus 1 etc which is equal to

1/2 and so zeta of 0 is equal to -1/2. Here just a couple more interesting

values for zeta. Nice. The zeroes are particularly interesting. In fact, it

turns out that zetas has zeros at all negative even integers, -2, -4,

-6, and so on. These are the so-called trivial zeroes of the zeta

function. I'm sure if you made it this far you've

also heard of the Riemann hypothesis which is all about other zeros of the

zeta function. It says that all other zeros are situated on this blue line

here and what's also really interesting is that on the right hand side we

actually don't have to super sum to calculate the values of zetas on the

blue line because just with ordinary summing eta can be evaluated

everywhere in this part of the complex plane, this part here which includes this

critical line. Lots and lots of other interesting things one could say here

but, well, we are here to wrap up this whole 1 plus 2 plus 3 is equal to

-1/12 business. Alright, ok, let's do it. In the first instance it is really our

identity up there that the Numberphile video is attempting to capture and it's

definitely tempting to express his identity as a new generalized sum

maybe like this. I've decorated the equal sign with an R in honour of Ramanujan who

seems to have been the first to think this way (NOT Euler as many people think).

In fact, in Ramanujan's notebook we can find a calculation very similar to the

one that's in the Numberphile video. Of course, there's a huge difference between

a monumental genius quickly abbreviating all his

complicated stuff in a personal note and a YouTube video addressed to a general

audience, right? What you see up there is part of a general method called

Ramanujan summation that assigns values to all sorts of divergent series

including the three Numberphile series. An important aspect of infinite series

which is often overlooked is the order in which the terms are summed. We are not

just adding an infinite set we are doing so in a certain order and this has all

sorts of important implications. So, for example, no matter how we arrange the

natural numbers into an infinite series this infinite series will always diverge

to plus infinity using standard summation. However, how exactly, how fast,

slow, regular, erratic it diverges to infinity depends very much on the order

of the terms in the series. The Ramanujan sum lacks pretty much all the nice

summy properties that we encountered today. However, it manages to capture

aspects of naturally ordered series and it pops up in many other branches of

the theory of divergent series in addition to the one we talked about

today. Check out some of the links in the description, especially if you know

a lot of maths the article by Terry Tao. Also I'm planning another video on the

so called Euler-Maclaurin summation formula which establishes a powerful

connection between sums and integrals and which is the starting point for

Ramanujan's sum. Just to whet your appetite here's one closely related

-1/12 fact. The nth partial sum of our infinite series is N(N+1)/2

So, if we plug in 1, 2, 3, 4 etc. for N the formula will spit out those partial

sums 1, 3, 6, 10. Now let's replace N by x and graph the resulting function. That's

a quadratic with zeros at 0 and - 1. Now the remarkable thing about this

graph is that the signed area here is equal to -1/12. And this is

definitely no coincidence. Really amazing stuff, don't you agree? And that's it,

finally, for today (and now I go and kill myself :) and I promise

the next video will be a LOT shorter.

For more infomation >> Numberphile v. Math: the truth about 1+2+3+...=-1/12 - Duration: 41:44.

-------------------------------------------

Golden milk recipe | "Golden milk" your life will change! - Duration: 5:05.

For more infomation >> Golden milk recipe | "Golden milk" your life will change! - Duration: 5:05.

-------------------------------------------

Top 10 Awesome Comic Book Supervillains Coming To Screens In 2018 - Duration: 6:44.

2017 was a good year for superheroes 2018 is

all poised to be even better movies like Avengers infinity war Black Panther and

Aquaman are already creating so much buzz with so many comic book movies

hitting the theaters this year let us have a look at all the supervillains our

favorite superheroes will be crossing paths with here top 10 awesome comic

book super villains coming to screens in 2018 10 Jessica Chastain Smith x-men

Dark Phoenix it is still unclear what role does Jessica Chastain actually

portray in the movie she is only credited as Smith a vaguely named

antagonist fans speculate she could be the Shire empress lilandra death bird or

another high-ranking member of her species others say she could be one of

the scrolls a shape-shifting species that has fought the Avengers several

times she is even thought to be the gender flipped version of mastermind

a mutant with vast psychic illusion and mind control abilities we won't know for

sure until the movie releases 9 Black Manta Ackerman Black Manta in the comics

is a renowned treasure hunter with an intense hatred for the king of the seven

seas his enmity with Aquaman has led him to cut off Aquaman's right arm as well

as kill his infant son Black Manta is also the father of a quality Aquaman's

sidekick powered by high-tech battle suit Black Manta will take on Arthur

curry in the movie eight black order Avengers infinity war for the first time

in the history of Hollywood the henchmen look cooler than the bad guy himself the

black order consists of four alien beings of pure hate and malice brought

from five corners of the universe and serve as tunnels de-facto generals in

this army they are Corvus glaive call obsidian Abunimah

and Proxima midnight seven claw black panther Andy Serkis is

awesome his acting is awesome his voice is awesome we'd get all three and much

more in black panther circus will portray ulysses klaue a renegade arms

dealer that killed tail his father and lost his hand in the process in the

comic books Klaw transforms into a being made of pure sound the movie still shows

circus in one piece so we have to assume Klaw will use his wits to take down

Black Panther in the movie six demon bear New Mutants nope it is not an 80s

slasher movie villain we are talking about here demon bear is an astral

entity that feeds our negative emotions in the comic books it even once gravely

injured Danielle Moonstar a prominent x-men with psychic abilities in a 1984

comic book arc demon bear is all set to appear in the New Mutants we hope demon

bears on screen appearance will also be as equally terrifying five ocean master

Aquaman ocean master is Aquaman's jealous

half-brother and the heir to the throne of Atlantis it is this jealousy towards

Ackerman that made the former the latter's most significant enemy till

date in the comics worm also known as ocean master wields a trident that gives

him weather and magic manipulating powers apart from his own Atlantean

physiology Patrick Wilson will portray norm in the movie for Brainiac Krypton

Brainiac is one of Superman's deadliest foes but until now

he was completely overlooked in all of Superman's movies and TV shows will not

anymore as Brainiac will become a part of sci-fi script on TV series brainiacs

predecessor role ducks will be featured in the TV show time-traveling

superheroes like Adam Strange and Hawkgirl are also set to appear three

Dark Phoenix x-men Dark Phoenix the Phoenix force is a pretty well known

cosmic entity in comic books it is the embodiment of all psychic power in the

universe and Jean Grey is its host the movie will take cues from the Phoenix

saga in the Dark Phoenix saga the x-men will move to space as Jean Grey loses

control of the Phoenix force as it turns her into something sinister expect

several other alien races to make their debut as well

- Eric kill monger black panther it is not always that we see a person of color

grab the spotlight in a Hollywood movie those that actually do end up becoming

superstars Michael B Jordan will assay the role of Eric hill monger a wakandan

exile that moves to America and becomes a genius and a proficient martial artist

kill hunger in the comic books has even assumed the role of Black Panther for a

while considering the trailer the movie is not going to show him in any

different light one fanos Avengers infinity war this guy is going to

freakin rock it Thanos is called the mad Titan for a

reason a renegade member of a genetically enhanced super species

called the Eternals fellow seeks absolute power to rule over the entire

universe he already has a terrifying reputation among several star systems

it's time for Thanos to live up to his name

you

For more infomation >> Top 10 Awesome Comic Book Supervillains Coming To Screens In 2018 - Duration: 6:44.

-------------------------------------------

U.S. State Department Issues Travel Warning To Mexico - Duration: 3:41.

The U.S. State Department has issued an urgent "do not travel" advisory warning to U.S.

citizens travelling to Mexico.

According to the warning, citizens are advised to avoid five Mexican states due to the violent

crime and gang activity.

USA Today reports: While the State Department has long recommended travelers exercise "increased

caution" in Mexico in general because of widespread homicide, kidnapping, carjacking

and robbery, the new warning elevates the five states to level 4, the highest level

of potential danger.

The advisory, issued Thursday, puts the states of Tamaulipas on the U.S. border and Sinaloa,

Colima, Michoacan and Guerrero on the Pacific coast on the same level as war-torn countries

like Syria, Yemen and Somalia.

The states have long been plagued with drug cartel activity like trafficking routes or

the cultivation of drug-related crops.

Turf wars between rival drug cartels have torn apart Tamaulipas, and Sinaloa is home

to the cartel of the same name.

Michoacan was so dominated by a drug cartel that vigilantes took up arms in 2013 to drive

them out.

Homicides skyrocketed in Colima in recent years due to the growth of the Jalisco New

Generation drug cartel, and the state now has Mexico's highest homicide rate, with

83.3 killings per 100,000 residents, according to figures from the first 11 months of 2017.

In addition to the level 4 warnings, the State Department singles out 11 states for a level

3 warning, which urges people to "reconsider travel."

Violence claimed 22,409 lives in Mexico in the first 11 months of 2017, the highest toll

since crime statistics began being released in 1997.

The State Department warning did not raise the current level 2 warning for Baja California

Sur or Quintana Roo, where two popular tourist destinations — Los Cabos and Cancun — are

located.

The level 2 advisory calls on travelers to exercise caution but does not explicitly suggest

they avoid the areas.

However, at least two Mexican resorts — Ixtapa-Zihuatanejo and Acapulco — are in a do-not-travel state,

Guerrero, and last year, the State Department extended a total ban on personal travel by

U.S. government personnel there.

The latest advisory threatens to further hurt Mexican tourism, a $20 billion industry accounting

for about 7% of the country's GDP.

The new advisory is the first for Mexico under the State Department's revamped system,

announced Wednesday, for issuing travel warnings.

While not addressing the latest warnings directly, the government's Mexico Tourism Board said

in a statement that "Mexico's major international tourism destinations have been explicitly

listed as having no travel restrictions," apparently a reference to major resorts like

Cancun, Puerto Vallarta and Huatulco.

In November, Mexico's Minister of Tourism, Enrique de la Madrid, walked a fine line in

acknowledging a crime problem in Mexico while trying to promote tourism.

"I'm very respectful of what the State Department has to do … what I'm saying

is those numbers aren't necessarily to be considered for tourism purposes because they're

describing (a different) situation," he told The Dallas Morning News.

"We just want to send a signal, which is true, that our destinations are a safe place

to visit."

For more infomation >> U.S. State Department Issues Travel Warning To Mexico - Duration: 3:41.

-------------------------------------------

8 Ball Pool - Free Coins + Box + Spin Reward For Free 3.12.4 Latest 2018 - Duration: 2:59.

For more infomation >> 8 Ball Pool - Free Coins + Box + Spin Reward For Free 3.12.4 Latest 2018 - Duration: 2:59.

-------------------------------------------

Medicaid Announces 2018 Changes, Millions On Welfare Affected - Duration: 2:11.

Trump is the President of accountability.

He wasn't elected to office to keep the status quo.

President Trump is going to clean up Washington.

That means fixing the massive government programs that drain taxpayer dollars.

Already he's done amazing work.

This year shapes up to be even bigger.

Reforms to government entitlements will mean fewer people are dependent on the state.

They will be empowered to work hard and earn their living.

That's just what the Trump administration will achieve.

Their big change to Medicaid looks to shake things up for good.

From Axios:

The Trump administration this morning took the first formal steps toward adding work

requirements to Medicaid — likely one of the biggest changes this administration will

make to the health care system…

Work requirements are a way to pare back Medicaid coverage, allowing states to remove some able-bodied

adults from their rolls.

In the face of congressional Republicans' inability to pass a larger health care bill,

these administrative actions are among conservatives' best chances to pull Medicaid in a more conservative

direction, after the Affordable Care Act dramatically expanded the program…

The Centers for Medicare & Medicaid Services says in a new guidance document that it "will

support state efforts to test incentives that make participation in work or other community

engagement a requirement for continued Medicaid eligibility."

A work requirement is a basic way to make sure people don't abuse government programs.

If someone is healthy enough to work, they should.

If they are getting special handouts by the government, surely they can meet basic requirements.

Democrats have been blocking such common sense changes for years.

They don't think Americans are capable of meeting these standards.

The result?

Millions of people dependent on the government.

We've created a virtual pauper state, thanks to programs like Welfare, food stamps, and

Medicaid.

But things are changing.

This is a small step in ensuring people will not abuse the government's generosity.

And it will empower people to find work, earn their own living, and support their families.

What's wrong with that?

For more infomation >> Medicaid Announces 2018 Changes, Millions On Welfare Affected - Duration: 2:11.

-------------------------------------------

소녀시대 멤버 탈퇴 정확한 이유는?|조회수4.989.283 - Duration: 3:42.

For more infomation >> 소녀시대 멤버 탈퇴 정확한 이유는?|조회수4.989.283 - Duration: 3:42.

-------------------------------------------

New Jurassic Park Mobile Command Center Lego Jurassic World 10 Hybrid Surprise Eggs Mutant Freaks - Duration: 11:44.

Wow guys keep watching as we fill these 10 hybrid mutant freak Jurassic world

Jurassic Park toys using the drastic park chaos of mobile Koreans

super excited today today we have 10 jillion eggs we're gonna open them up

and build 10 Jurassic world hybrid mutant dinosaur freaks it's gonna be

New Jurassic Park Mobile Command Center Lego Jurassic World 10 Hybrid Surprise Eggs Mutant Freaks

mobile command centers we let's get started okay let's go ahead and check

out our first tag awesome Wow what are we gonna build with yes

okay so here is our first mutant hybrid freak Wow what in the world is this guy

supposed to be I have no idea if you want to come up with a name for him I

would say some type of a Moses or type creature but okay let us check out our

next

what are you gonna make this

okay so here is our mutant freak huge legs and he's got huge legs for arms and

he's got wings Wow I wouldn't want to meet this guy

anywhere okay and our third one is

why

hey slow

weird weird weird our third mutant freak we're gonna set him up here in the

trailer if he could stay

okay and our fourth one

Wow that's a lot of weird-looking parts

this guys supposed to run forward or sideways is the herbivore or a carnivore

he has a carnivore body but herbivore heads everywhere two huge legs three

Triceratops heads and wings walk I feel sorry for some of these get okay for our

fifth hybrid

okay so here is our next one what and the world is big indominus rex arms big

legs small Dilophosaurus body big t-rex head with a little indominus rex inside

sort of like if you ever seen aliens he's got the little head popping out

that is this guy hey r6 hi fridge freak we have wow I have no idea what we have

chick

okay so little Triceratops body big Triceratops head small t-rex arm big

indominus rex arm velociraptor head and a pterodactyl head and wings so this

guy's got do it okay that is enough to set these guys up better offer

okay that is a lot of parts for one egg

hey let's check out this poor unfortunate guy t-rex body Triceratops

but big indominus rex arms for these two arms he's got Triceratops body for this

one he's got a little indominus rex body with big indominus rex arms and he's got

wings so i have no idea which way this guy is supposed to go

yeah this guy looks good using he's got a leg coming off his butt Triceratops

body and ominous Rex legs indominus rex and t-rex tails for arms

you've got indominus rex head with a t-rex head inside and little tons of

things back

who comes up with these ideas

okay so t-rex body pig Triceratops head - truck indominus rex and t-rex hands

forearms he's got wings and tiny little arms four legs so how this guy travels

okay and our final one is Wow who comes up with these crazy ideas man this guy

must be totally oops that's me

okay huge t-rex body t-rex arms four legs wings you've got two tails four

arms Dominus Rex head with a tiny little

t-rex head inside like again from the movie Iliad ok guys so here is are all

10 of our hybrid mutant 3 dinosaurs in our Jurassic Park chaos effect mobile

command center they are totally awesome if you guys enjoyed this video you want

to see more fun videos go ahead click the subscribe button also click the bell

button next to it to be notified when I make videos do have over a thousand

videos guys majority are Jurassic world Jurassic Park

Godzilla King Kong in Power Rangers Wow that was totally awesome you guys are

the best click the subscribe button below for a lot more fun video also

click the bell button to be notified every time I make a new video click the

boxes below for a lot more fun videos and if you want to see even more go

ahead and click the subscribe button

For more infomation >> New Jurassic Park Mobile Command Center Lego Jurassic World 10 Hybrid Surprise Eggs Mutant Freaks - Duration: 11:44.

-------------------------------------------

The Hidden Secrets Built Into Medieval Castles That Most People Never Knew Existed - Duration: 6:03.

when you look at medieval castle several different images probably pop into your

head it could be of a princess longingly gazing out of a window or perhaps a

knight jousting atop is noble white steed

we all hold different associations with castles that's certainly true but how

much do we really know about their design and a purpose they serve for

those living behind the walls while a mediaeval castles might seem arcane they

did serve a very specific function when they were built they also hold dark

secrets the first time you probably heard of a castle was when someone was

reading fairytales before bed perhaps you pictured him as the sparkling

mansions where Cinderella and the other princesses found their happily ever

afters the truth is the castles are all embellished with gold and gems in

reality when castles were erected during the medieval era they served the very

specific purpose to defend and protect the people behind the walls

today's castles might be museums or houses for royalty but when the original

medieval castles were built they were designed to serve as fortresses during

times of war all of the planning that went into them was about defending the

grounds from enemies when you think of a castles first line of defense you

probably imagine a moat right traditionally a moat was a large body of

water that circled the castle and separated it from the land

but it wasn't there to keep enemies from crossing for people designing most

castles their biggest fear was that their enemies might dig underneath the

walls to gain entry if there was a moat it ensured that any tunnels would be

immediately flooded in fact for some castles the moat wasn't even located

outside the place at all instead it could be found between the first and

second walls of the castle the way anyone digging a tunnel would get a

truly unpleasant surprise speaking of unpleasant surprises modes

didn't exist elite to keep invaders from digging tunnels to gain entry either the

moat served other purposes for those living in the castle

for instance they made for a great way to dispose of human waste when it came

to other methods of protecting the castle one of the oldest traditions in

design was the concentric circles of defense looking at this castle from

above you can see how the circles were created to make entry very difficult

concentric circle defense were designed to act as a series of obstacles while

the layout of medieval castles might be all too familiar to us when it comes to

how we look at castles today their true innovation in the world of design when

they were originally built the concentric circles of defense meant that

invading armies would have to conquer one obstacle after another slowing them

down as they made their assault on the castle first there was a wall then there

was a moat then another wall a keep and so forth

the main gauge of the castle might look imposing but to our modern eyes that's

all it is the truth of the matter is that during medieval times the main gate

of the castle was more than intimidating it was downright deadly the main gate

was often comprised to two barriers if invaders made it past the first entryway

they could become trapped between the first to the second gate by castles

inhabit they might think that their siege was

going to be successful but they were wrong the invading soldiers would n be

trapped in one of the castle courtyards and this wasn't pleasant to say the

least they were thin slits in the courtyard walls that had allowed in a

castle inhabitants to fire upon the trapped intruders the secret

fortifications of castles doesn't end there either

even the stairwells played an important role most of them were designed to be

deliberately narrow and to spiral up in a clockwise fashion and there was a very

sneaky reason why they were designed of this way so the soldiers running up them

would have their sword hands usually their right against the wall the narrow

and curving stairs would make it that much harder for them to reach for their

weapons of course the people who lived inside of the castle would know the

layout of the space very well right down to the crowded cramped and narrow

staircases this gave them the ultimate advantage schitot reuters breached the

walls secret passages were a critical part of the design of any and every

castle that was built these passages could serve many different purposes

including allowing a means of escape for those who lived in the castle these

secret passages could also lead to rooms where the castles inhabitants could take

shelter in the event of a siege they can also serve as a great way of getting

much-needed supplies and other assistance into the castle sometimes the

secret pass has led to secondary wells for the castles inhabitants in case the

attackers breached their walls of defense and poisoned their drinking

supplies castles may be glamorous but they were also critical war fortresses

that's just the tip of the iceberg when it comes to all of the secrets of the

way meaty evil castles were made some of these designs were truly genius

the design and forethought that went into these Horace's was really something

else who knew the castles like these were so complex

For more infomation >> The Hidden Secrets Built Into Medieval Castles That Most People Never Knew Existed - Duration: 6:03.

-------------------------------------------

Nightcore - All These Years (Camila Cabello) - Lyrics - Duration: 3:12.

This video includes lyrics on the screen

For more infomation >> Nightcore - All These Years (Camila Cabello) - Lyrics - Duration: 3:12.

-------------------------------------------

BREAKING: Feral Thugs STEAL And Run Over Elderly White Man- 1 Captured, 3 More On The Run - Duration: 3:45.

Four black thugs in Baltimore County have been arrested after police allege they brutally

attacked an elderly man and drove over him with his own car before they stole it.

What is wrong with these people?

It's like most of these Negroes have no brains and no soul.

They are evil trash.

The thugs — Darian Maurice Hicks, 16; Taizjuan Eliza Waddell, 16; and Kimani William Graham,

16 — were arrested earlier this week by authorities in Baltimore, WJZ-TV reported.

The victim, identified as 69-year-old Jim Willinghan, was "violently attacked" around

10:30 a.m. on Jan. 3, according to The Baltimore Sun.

Willinghan reportedly told police he was getting something from the trunk of his Buick LaSabre

when two cars sped past him.

The two vehicles then allegedly turned around and sped toward him, before the useless black

thugs got out of their cars to threaten him.

James1 Jim Willinghan, 69, was hospitalized after

police say he was attacked by six teenagers in Baltimore County last week.

He was left with several injuries, including a broken pelvis.

(GOFUNDME)

The black trash, according to charging documents, threatened to shoot Willinghan as they took

his wallet and keys.

One of the teens then got into Willinghan's car, and reversed it back into him — reportedly

driving over his right arm and hand.

Fu*k these assholes.

The teens then allegedly drove off and left Willinghan, who's now recovering at the

R Adams Cowley Shock Trauma Center, with a broken pelvis, Fox News reports.

"I just don't understand the reasoning," wife Barbara Willinghan told WMAR-TV of the

attack.

"I mean my husband complied.

Why would you want to beat the poor man half to death.

The detective said he's lucky to be alive."

Because these thugs are useless trash and belong in cages.

"It's going to be a slow-going process.

My husband can't walk right now, and he's got a big bruise on his back, slight internal

bleeding," Willinghan's wife added in an interview with WJZ-TV.

This is just another reason why we all need to be armed.

If these assholes did this to me they would be full of lead.

A Comcast employee who overheard the incident reportedly recorded it and handed it over

to investigators.

The employee allegedly watched the suspects as they got into their cars and drove off,

and followed them to an area of the city where police eventually arrested them, The Baltimore

Sun reported.

Police recovered the victim's Buick, and two other vehicles.

A GoFundMe page for the Willinghans has been created "to help alleviate some of the financial

burden caused by this horrific incident," as Jim, with multiple injuries, "is the

sole provider for their home."

Hicks — identified as the teen who drove over Willinghan — was charged with attempted

second-degree murder, in addition to a number of other charges.

Waddell and Graham have both been charged with armed carjacking, carjacking, armed robbery,

robbery, first-degree assault, second-degree assault, unlawful taking of a motor vehicle,

and theft charges.

A 15-year-old suspect was also arrested in the case and charged as a juvenile.

Law enforcement officials are still searching for two more suspects involved the attack.

Look, these Negroes are obviously useless- can't we just execute them on the spot?

Why pay millions to keep assholes like this alive?

That goes for any race- but this article is about Negroes attacking a white person.

We are at war.

We will win.

There is no time or money to waste on these filthy savages.

Feel free to comment below.

For more infomation >> BREAKING: Feral Thugs STEAL And Run Over Elderly White Man- 1 Captured, 3 More On The Run - Duration: 3:45.

-------------------------------------------

President Trump BASHED Liberals In A PATRIOTIC WAY! – Way To Go! - Duration: 1:56.

This morning President Trump absolutely crushed the liberal immigration policy with a strong

patriotic statement that he laid down on Twitter.

In his five-tweet statement, he stunned the liberal public and made them not to know where

to turn, since they all know about their leader's corruption, only they didn't want to say

it.

POTUS bravely tackled them saying:

"The so-called bipartisan DACA deal presented yesterday to myself and a group of Republican

Senators and Congressmen was a big step backwards.

Wall was not properly funded, Chain & Lottery were made worse and USA would be forced to

take large numbers of people from high crime….."

"….countries which are doing badly.

I want a merit based system of immigration and people who will help take our country

to the next level.

I want safety and security for our people.

I want to stop the massive inflow of drugs.

I want to fund our military, not do a Dem defund…."

"….Because of the Democrats not being interested in life and safety, DACA has now

taken a big step backward.

The Dems will threaten "shutdown," but what they are really doing is shutting down

our military, at a time we need it most.

Get smart, MAKE AMERICA GREAT AGAIN!"

"The language used by me at the DACA meeting was tough, but this was not the language used.

What was really tough was the outlandish proposal made – a big setback for DACA!"

"Sadly, Democrats want to stop paying our troops and government workers in order to

give a sweetheart deal, not a fair deal, for DACA.

Take care of our Military, and our Country, FIRST!"

The President really showed them how things are around here.

He is 100% true and 100% Patriot, a deadly combination for the anti-military, anti-flag,

anti-anthem, anti-America liberals who simply cannot escape from the reality.

For more infomation >> President Trump BASHED Liberals In A PATRIOTIC WAY! – Way To Go! - Duration: 1:56.

-------------------------------------------

THE KIRA JUSTICE - "Eclipse" (Música Original escrita durante LIVE │ TV size) - Duration: 2:04.

For more infomation >> THE KIRA JUSTICE - "Eclipse" (Música Original escrita durante LIVE │ TV size) - Duration: 2:04.

-------------------------------------------

Everymonth 7-8 thousand dollar earning in online!! - Duration: 9:51.

Everymonth 7-8 thousand dollar earning in online

For more infomation >> Everymonth 7-8 thousand dollar earning in online!! - Duration: 9:51.

-------------------------------------------

Maxima & Willem-Alexander: Furchtbare Ehe-Krise - Duration: 4:48.

For more infomation >> Maxima & Willem-Alexander: Furchtbare Ehe-Krise - Duration: 4:48.

-------------------------------------------

Fritz Wepper: Angst um sein Leben! - Duration: 4:00.

For more infomation >> Fritz Wepper: Angst um sein Leben! - Duration: 4:00.

-------------------------------------------

গুগল পিন ভেরিফিকেশন ২০১৮।How to send google adsense pin verification letter to your address in 2018. - Duration: 4:00.

Subscribe my chanel for updates

For more infomation >> গুগল পিন ভেরিফিকেশন ২০১৮।How to send google adsense pin verification letter to your address in 2018. - Duration: 4:00.

-------------------------------------------

과거를 용서하고 놓아주어라 - Duration: 7:20.

For more infomation >> 과거를 용서하고 놓아주어라 - Duration: 7:20.

-------------------------------------------

Genus Ordinis Dei - "Cold Water" (Making of the Music Video) - Duration: 9:25.

Hi guys!

Fassa Valley, more specifically Pordoi Pass.

We are above 2000 meters and we are shooting the video of Cold Water.

We got some good for you!

This is just the first (and only) part of this making of...

so, follow us and stay tuned!

I think it was better with the drone...

you sure about it?!

Guys, we are in Canazei where you can party hard with Genus Ordinis Dei.

The situation after 2AM is the following...

Mr. Steve Saints (Director) on your left (and mine too).

Tommy in the center and Steven on the bass.

They are going to sleep on the old beds of me and my brothers.

Take a look at those classy blankets

Mr. Richard Meiz on the Drums.

Obviously this video is also for our girlfriends ...

to let them know the level of sexual content.

Tomorrow it will be very hard

Where we will basically go to

Sella Pass or Pordoi Pass or whatever...

and we'll shoot a fenomenal video.

Steven, what are you reading?

eeeehmm... "The World at the Edge of the World"

Sorry! "... at the end of the World".

and you had to read it on the cover?!

Brando? (aka Tommy)

Brando, What about you?

The same shit over the last 20 years...

Tolkien!

Tolkien, guys!

'Night.

Goodnight, guys!

Here will sleep yours truly with The Drummer.

See you tomorrow! Cheers!

Well, 7:30AM

and this is the situation at Nick's house.

We are really excited about climbing the Sella and Pordoi pass.

to shoot this super secret ...

You can see some military jackets on the background...

try to guess why...

Nick, if you say: "Guess why.." in a "Making of" you are NOT creating hype

Because the "Making of" is coming out AFTER the video.

So, they all already know what the jackets were for.

You are not stupid, my friend.

That's why I picked you up in the Band in 2012.

You'd better learn how to make live streams on Facebook, @ss#ole!

Here we are...

Sella Pass,

Pordoi Pass,

Sa... Saslong

Chaise long it's a sofa!

Chateau d' Ax...

F**K this s#it! You actually don't know mountains.

I used to go to Sasso Lungo! (Saslong)

Really?!

Yes, to take it in the @ss!

Cheers, guys! Say "Hello" to the camera!

Please Meiz, not like a pussy!

Tommy?!

This is the situation!

Our destination.

What a place!

Over there?!

Exactly!

Steven, show us the way!

Ok... why me?!

Guys...

#QUALITY

YEEESSSSSSSSS!!!!

Say "Hello" guys!

Are you f***in kidding me?!

YEEEEHH!!

Guys, This is pure awesomeness...

We recorded some drone scenes

and I think that

Spielberg,

Nolan

and the other motherf****rs

will be jealous for the next three years.

The problem is those three pussies

who are always stopping every 30 meters.

But take a look to the landscape!

Really amazing!

3...

2...

No, it was just for the Backstage... was funny!

Action!

He did the "Prosciutto Face"!

C'mon, c'mon, COME ON!

It's f***in cold, Meiz!

Action!

*FAIL.

Stop.

We are coming!

Mr. Steve Saints is slow as F**K!

I'm suffering more than you!

Next time we gonna shoot "HOT WATER"!

and It will be on a SAUNA!

*°ÇÇ∞ZçéP!

We ready!

Guys! It's a f***n BOMB!

Ready? Go!

Look at this!

Let's go to the guy's!

This is great, bro!

- Let's cook the chicken! -

He was afraid to go higher! You bastard!

What do you think about this calibration?

I think that it has been done in a proper way.

Cheers to the Camera!

GUYS?!

Say "Hello"!

F**K YOU!

Guys...

I really hope...

...that you'll understand...

...the situation, with this video.

...But it will be even more brutal.

What the F**K are you doing here?!

Can you please f**k out of here?!

C'mon, get the f**k out!

...I was just playing down!

Steve Saint, I have an idea...

I'd say: super high quality, in slow motion...

But there's someone on the background who act like this!

People would get mad about it.

I mean...just like this!

Get the F**K out!

GET OUT!!!!

C'MON!!!!

Bye.

No, It's just to study the photography...

Yeah, sure...

C'mon, go!

You are REALLY an idiot...

We gonna do a camera-check later...

Are you recording, or are you NOT recording?!

I am.

Not all...just a little piece...

How do I grab it?

With your hands!

C'mon, do it!

It taste like meat.

Put some more!

No, NO! it's enough!

No more...it's too much!

It may look fake!

Ok, we just finished the cannibalism scene!

- I KILL YOU! -

Nick?!

You are a piece of s#it!

Oh, actually, I AM the piece of s#it

Because I decided to do this video!

WHAT?!

So YOU are the piece of s#it?!

Get dressed and stop complaining!

Really?

Comments?

I hate you! Get in this f***in car!

The best actor is the corpse.

Steven?

!&%$[]##@!!!

Ok guys, with this view...

We finished the recordings.

I'll show you our cave!

The one in the center, as you can see!

Yes

Yes

Yes

Bro - ther!

So?

So what?

What do you think about this video?

It's a BOMB!

...A bomb...