# Proving ∏ with summation and/or Lhopitals?

1. Apr 30, 2012

### LearninDaMath

1. The problem statement, all variables and given/known data

Someone in school was showing me this proof or problem that, I believe, proves or yields π via this limit: Lim x-->0 of $\frac{xπcot(πx)}{x}$-$\frac{1}{x}$ = tan(0) = 0

And that this somehow related to a summation $\sum1/k^{2}$ as the sum goes from 1 to ∞.

I don't recall any of the details about the proof or what the proof is proving (but I think it has something to do with proving that that π=3.14 as the sum gets closer to infinity.)

Does anything like this look or sound familiar to any known proof?

Last edited: Apr 30, 2012
2. May 11, 2012

### Dustinsfl

There seems to be some confusion with what you have.

First $\sum\limits_{n=1}^{\infty}\frac{1}{n^2}=\frac{ \pi ^2}{6}$.

Here is the proof. I will be using the series for $(\pi\cot\pi z)'$ at $z=0$

We have that $\sin\pi z = \pi z\prod\limits_{n = 1}^{\infty}\left(1 - \frac{z^2}{n^2}\right)$ which is an entire function with simple zeros exactly at the integers.
We can differentiate termwise by uniform convergence.
So by logarithmic differentiation, we obtain a series for $\pi\cot\pi z$.
$$\frac{d}{dz}\ln\left(\sin\pi z\right) = \pi\cot\pi z = \frac{1}{z} - 2z\sum_{n = 1}^{\infty}\frac{1}{n^2 - z^2}$$
So we have that $-\sum\limits_{n = 1}^{\infty}\dfrac{1}{n^2 - z^2} = \dfrac{\pi\cot\pi z - \frac{1}{z}}{2z}$.
In order to observe that the limit as $z$ goes to 0 works, we need to look at the power series of
$$\pi\cot\pi z = \frac{1}{z} - \frac{\pi^2}{3}z - \frac{\pi^4}{45}z^3 -\cdots.$$
So by taking the Taylor series expansion of $\cot\pi z$ and multiplying by $\pi$, we have
\begin{align}
\frac{\pi\cot\pi z - \frac{1}{z}}{2z} &= \frac{- \frac{\pi^2}{3}z - \frac{\pi^4}{45}z^3 -\cdots}{2z}\notag\\
&= -\frac{\pi^2}{6} - \frac{\pi^4}{90}z^2 -\cdots.\notag
\end{align}
At this point, the singularity at 0 has been removed.
So
\begin{align}
\lim_{z\to 0}-\sum_{n = 1}^{\infty}\frac{1}{n^2 - z^2} = \lim_{z\to 0}\frac{\pi\cot\pi z - \frac{1}{z}}{2z}\notag\\
-\sum_{n = 1}^{\infty}\frac{1}{n^2} = \lim_{z\to 0}\left[\frac{\pi\cot\pi z - \frac{1}{z}}{2z}\right]\notag\\
\sum_{n = 1}^{\infty}\frac{1}{n^2} = \frac{\pi^2}{6}.\notag
\end{align}

Last edited: May 11, 2012
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