Sum converging to pi^2/6, why?

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[SOLVED] Sum converging to pi^2/6, why?

I've seen the identity,
[tex]\frac{\pi^2}{6}=\sum_{n=1}^\infty \frac{1}{n^2}[/tex]
but I've never seen a proof of this. Could anyone tell me why this is true?
 
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  • #2
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well this is a p-series. So the series [tex] \sum_{n=1}^\infty \frac{1}{n^2}[/tex] , converges if

[tex]\int_1^{\infty}\frac{dx}{x^{2}}=\lim_{b\rightarrow\infty}\int_1^b x^{-2}dx=-\lim_{b\rightarrow\infty}( \frac{1}{x}|_1^b)=-\lim_{b\rightarrow\infty}(\frac{1}{b}-1)=1[/tex]

I don't see how would this converge to what u wrote though.. sorry..
 
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  • #3
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I'm well aware that the sum converges, but I'm curious why it converges to [itex]\frac{\pi^1}{6}[/itex], and not some other value. The integral and the sum have different values. The integral is 1, the sum is not.
 
  • #4
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I'm well aware that the sum converges, but I'm curious why it converges to [itex]\frac{\pi^1}{6}[/itex], and not some other value. The integral and the sum have different values. The integral is 1, the sum is not.

Have you learned power series?? I think you have to write the power series for that serie, and after that use the methods of power series to calculate that sum.
 
  • #5
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Have you learned power series?? I think you have to write the power series for that serie, and after that use the methods of power series to calculate that sum.
If this were a power series, it would involve [itex]n![/itex].
 
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  • #9
That's a very nice explanation, Euler was a true master of mathematics :smile:
 
  • #10
HallsofIvy
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Have you learned power series?? I think you have to write the power series for that serie, and after that use the methods of power series to calculate that sum.
The power series for a series? You can think of a numerical series as a power series (in x) evaluated at a specific value of a but there are an infinite number of power series that can produce a given series in that way.

If this were a power series, it would involve [itex]n![/itex].
No, a power series is any series of the form [itex]\Sum a_n x^n[/itex] where [itex]a_n[/itex] is any sequence of numbers. Even the Taylor's series for ln(x) does not involve n!
 
  • #11
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No, a power series is any series of the form [itex]\Sum a_n x^n[/itex] where [itex]a_n[/itex] is any sequence of numbers. Even the Taylor's series for ln(x) does not involve n!
My bad. Most involve n!, but all involve n.
 
  • #12
HallsofIvy
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My bad. Most involve n!, but all involve n.
Except those that involve "i"! Do you have any support for your statement that "most" power series involve a factorial? That is certainly not my experience.
 
  • #13
CRGreathouse
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Except those that involve "i"! Do you have any support for your statement that "most" power series involve a factorial? That is certainly not my experience.
gamesguru just took a weighted average over all power series, giving ones he (?) knew weight 1/n and all others weight 0. :wink:
 
  • #14
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e^x (hyperbolic functions included), sin[x], cos[x], tan[x] all have a factorial in their power series. The only useful examples I can think of that don't have a factorial are the inverse trig functions and the natural log. Anyways, I don't want to get into an argument, I'll just rephrase myself, most power series that I've seen and observe as useful, involve a factorial. And no, I can't prove that.
 

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