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 2 years ago
This shouldn't be too hard (for the peeps, here, at least):
Give an analytical, definite value for the sum of the entire set \(\mathbb{N}\) starting from its lowest element and transversing through it in order.
Then, prove it.
 2 years ago
This shouldn't be too hard (for the peeps, here, at least): Give an analytical, definite value for the sum of the entire set \(\mathbb{N}\) starting from its lowest element and transversing through it in order. Then, prove it.

This Question is Closed

LolWolf
 2 years ago
Best ResponseYou've already chosen the best response.1Hint: Think of the Riemann Zeta function.

dumbcow
 2 years ago
Best ResponseYou've already chosen the best response.0umm the infinite sum of the natural numbers is infinity

LolWolf
 2 years ago
Best ResponseYou've already chosen the best response.1Nope, sorry. I was asking for the analytical sum.

dumbcow
 2 years ago
Best ResponseYou've already chosen the best response.0sum of an arithmetic series \[S_n = \frac{n(a_1 +a_n)}{2}\] for set of natural numbers \[S_n = \lim_{n \rightarrow \infty}\frac{n(n1)}{2}\] which obviously goes to infinity i guess im not understanding what you're looking for

LolWolf
 2 years ago
Best ResponseYou've already chosen the best response.1Yes, the sum diverges. The question is not such, I'm asking for an analytical, total sum of the set of natural numbers.

dumbcow
 2 years ago
Best ResponseYou've already chosen the best response.0hmm i guess i never learned this or remember it...analyzing the cardinality of the sum in terms of infinity? or something like that

dumbcow
 2 years ago
Best ResponseYou've already chosen the best response.0my math knowledge stops at number theory and advanced analysis :

LolWolf
 2 years ago
Best ResponseYou've already chosen the best response.1Remember that when you use the method of taking \(n\to\infty\) and \(\frac{n(n+1)}{2}\) you are receiving a partial sum. Such will always diverge, and thus end up at infinity. And, nah, it's not morphisms for countability (i.e. \(\mathbb{Q^+}\) is countable by \(\mathbb{N}\))

ParthKohli
 2 years ago
Best ResponseYou've already chosen the best response.0\[{n \over 2}\left(a_1 + a_n \right)\]So here\[{\lim_{n \to \infty} {n \over 2}(1 + a_n)}\]Though, the last natural number can NOT be determined.

LolWolf
 2 years ago
Best ResponseYou've already chosen the best response.1Nope, it's an actual discreet value. Again, you can't use the \(\frac{n(n+1)}{2}\), because those are partial sums.

mukushla
 2 years ago
Best ResponseYou've already chosen the best response.0lol...i mean i would like to know what is the answer :)

LolWolf
 2 years ago
Best ResponseYou've already chosen the best response.1Hint 2: The series is not AbelSummable, but can be manipulated to do such, and manipulated back.

LolWolf
 2 years ago
Best ResponseYou've already chosen the best response.1@mukushla Oh, all right, haha, if no one gets it by tomorrow I'll post up the answer.

mathslover
 2 years ago
Best ResponseYou've already chosen the best response.0Hint 3 : try to search on google ... if no then follow mukushla's spirit :P

LolWolf
 2 years ago
Best ResponseYou've already chosen the best response.1(Although, I AM surprised no one has gotten this... how many of you have taken functional analysis/analytic class numbers?)

Compassionate
 2 years ago
Best ResponseYou've already chosen the best response.0*Watches you guys debate while eating cereal.* Lolgoodguessbrobutstillwrong.jpg

Compassionate
 2 years ago
Best ResponseYou've already chosen the best response.0This is boring. Prove it as N approaches infinity and put it in a summation, then use the formula stated above. Toodles.

LolWolf
 2 years ago
Best ResponseYou've already chosen the best response.1@Compassionate Lol, if it was that easy, don't you think that people would have gotten it within the first few minutes? Try doing that, yourself, see if the result is finite.

Compassionate
 2 years ago
Best ResponseYou've already chosen the best response.0RolloverlolnobackspacedeletethatCTRLALDELnopewrong.avi

LolWolf
 2 years ago
Best ResponseYou've already chosen the best response.1That would be for whose post...?

LolWolf
 2 years ago
Best ResponseYou've already chosen the best response.1All right, well, here's the answer. It might seem a little "outthere" for the firsttime reader (I would say it would even look almost magical), but it's not that imaginative, once you're used to these manipulations: Starting from the geometric sum \(S(x)\): \[\begin{align} S(x)&=1+x+x^2+\cdots\implies\\ xS(x)&=x+x^2+x^3+\cdots\implies\\ 1+xS(x)&=1+x+x^2+\cdots=S(x)\implies\\ S(x)&=\frac{1}{1x}=1+x+x^2+\cdots \end{align} \]All right, so, we take the derivative (this is one of those apparently 'magical' parts, which becomes pretty mundane after a while). If you've studied generating functions, chances are this is nothing new: \[ S'(x)=\frac{1}{(x1)^2}=0+1+2x+3x^2\cdots \]But, we can't get a definite sum for this series, since \(S'(x)\) has a pole at \(x=1\). So, with some foresight, we plug in \(x=1\) (it will become obvious as of why, later): \[ S'(1)=\frac{1}{4}=12+34+\cdots \](This was the second hint I gave, about Abelsummability. This shouldn't have to be derived, I'm just doing it as a little extra). All right, so, we now use the function \(\zeta(s)\) as a manipulation aid: \[ \zeta(s)=\sum_{n=1}^\infty n^{s} \]We wish to negate every even term of the series, to be able to solve back for our own, final value, so we begin: \[ \begin{align} \zeta(s)&=1^{s}+2^{s}+3^{s}+\cdots\implies\\ (12^{s})\zeta(s)&=1^{s}+2^{s}2^{s}+3^{s}+\cdots=1^{s}+0+3^{s}+\cdots\implies\\ (14^{s})\zeta(s)&=1^{s}2^{s}+3^{s}+\cdots \end{align} \]Which is what we wished to find. Now, we know the value of this last series manipulation at \(s=1\), so we use this to find our value for \(\zeta(1)\), which is the sum of the natural set: \[\begin{align} (14^{1})\zeta(1)&=12+34+\cdots=\frac{1}{4}\\ (3)\zeta(1)&=\frac{1}{4}\\ \zeta(1)&=\frac{1}{12} \end{align} \]So... guess what? The sum of all natural number is \(\frac{1}{12}\). Enjoy! http://en.wikipedia.org/wiki/1_%2B_2_%2B_3_%2B_4_%2B_%C2%B7_%C2%B7_%C2%B7

LolWolf
 2 years ago
Best ResponseYou've already chosen the best response.1Well, the answer's up: @dumbcow, @mukushla, @mathslover, @ParthKohli, @Compassionate

LolWolf
 2 years ago
Best ResponseYou've already chosen the best response.1Oh, sorry, I had a slight error on one part, but it's an easy correction and still gives the same result, change \[ (14^{s})\text{ to }(12\cdot2^{s}) \]

dumbcow
 2 years ago
Best ResponseYou've already chosen the best response.0yes never took these classes...but you are saying the sum of all positive integers equals a negative fraction huh

LolWolf
 2 years ago
Best ResponseYou've already chosen the best response.1I know, right? I said the same thing, haha, it's kinda ridiculous, but it's true.
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