Why Does the Sum Converge to π²/6?

[gamesguru]

[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?

 

[sutupidmath]

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..

 

[gamesguru]

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.

 

[sutupidmath]

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.

 

[gamesguru]

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 $n!$.

 

[ObsessiveMathsFreak]

Read Euler. He is the master of us all.

 

[nicksauce]

http://www.secamlocal.ex.ac.uk/people/staff/rjchapma/etc/zeta2.pdf

 

[gamesguru]

Read Euler. He is the master of us all.

Thanks that's what I wanted to see.

 

[Pere Callahan]

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

 

[HallsofIvy]

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!

 

[gamesguru]

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]

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]

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.

 

[gamesguru]

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.