- Thread starter
- #1

- Jan 17, 2013

- 1,667

Let us see how many different methods can we get

- Thread starter ZaidAlyafey
- Start date

- Thread starter
- #1

- Jan 17, 2013

- 1,667

Let us see how many different methods can we get

- Thread starter
- #2

- Jan 17, 2013

- 1,667

But there are other ways

Fourier series , Complex analysis

- Thread starter
- #3

- Jan 17, 2013

- 1,667

This requires the function to be analytic on the real axis but \(\displaystyle \frac{1}{k^2}\) has a pole of order \(\displaystyle 2\) at the origin .

So we can adjust the theorem to solve for poles on the real axis

\(\displaystyle \sum_{k\leq -1} \frac{1}{k^2} +\sum_{k \geq 1} \frac{1}{k^2}= - \text{Res}\, \left(\frac{\pi \cot(\pi z)}{z^2};0 \right)\)

\(\displaystyle \sum_{k\geq 1} \frac{1}{k^2} +\sum_{k\geq 1} \frac{1}{k^2}= \frac{\pi^2}{3}\)

\(\displaystyle \sum_{k\geq 1} \frac{1}{k^2} = \frac{\pi^2}{6}\)

I deleted the proof of the modification of the theorem above . If someone is interested I will try to post it.