# Sum converging to pi^2/6, why?

## Main Question or Discussion Point

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

I've seen the identity,
$$\frac{\pi^2}{6}=\sum_{n=1}^\infty \frac{1}{n^2}$$
but I've never seen a proof of this. Could anyone tell me why this is true?

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## Answers and Replies

well this is a p-series. So the series $$\sum_{n=1}^\infty \frac{1}{n^2}$$ , converges if

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

I don't see how would this converge to what u wrote though.. sorry..

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I'm well aware that the sum converges, but I'm curious why it converges to $\frac{\pi^1}{6}$, and not some other value. The integral and the sum have different values. The integral is 1, the sum is not.

I'm well aware that the sum converges, but I'm curious why it converges to $\frac{\pi^1}{6}$, 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.

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 $n!$.

Read Euler. He is the master of us all.
Thanks that's what I wanted to see.

That's a very nice explanation, Euler was a true master of mathematics

HallsofIvy
Science Advisor
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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 $n!$.
No, a power series is any series of the form $\Sum a_n x^n$ where $a_n$ is any sequence of numbers. Even the Taylor's series for ln(x) does not involve n!

No, a power series is any series of the form $\Sum a_n x^n$ where $a_n$ 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.

HallsofIvy
Science Advisor
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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.

CRGreathouse
Science Advisor
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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.

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.