Deriving Sum from Fourier Series

• Apr 4th 2011, 06:43 AM
StaryNight
Deriving Sum from Fourier Series
I've worked out the following Fourier series for exp(x) (valid between -pi and pi) and would now like to be able to derive the sum of 1/(1-n^2).

$e^x = \frac{1}{\pi} sinh(\pi) + \sum_{1}^{\infty} \frac{2sinh(\pi)}{\pi(1+n^2)} (-1)^n(cos(nx)-nsin(nx))$

This would work perfectly if I could substitute pi or -pi into the series, but unfortunately it is not valid for these values so gives the wrong result. Subbing in 0 works but this gives the alternating series (-1)^n/(1-n^2).

Can anyone give me a push in the right direction? I've tried differentiating/integrating the series but this does not seem to be helpful.

• Apr 4th 2011, 07:13 AM
TheEmptySet
Quote:

Originally Posted by StaryNight
I've worked out the following Fourier series for exp(x) (valid between -pi and pi) and would now like to be able to derive the sum of 1/(1-n^2).

$e^x = \frac{1}{\pi} sinh(\pi) + \sum_{1}^{\infty} \frac{2sinh(\pi)}{\pi(1+n^2)} (-1)^n(cos(nx)-nsin(nx))$

This would work perfectly if I could substitute pi or -pi into the series, but unfortunately it is not valid for these values so gives the wrong result. Subbing in 0 works but this gives the alternating series (-1)^n/(1-n^2).

Can anyone give me a push in the right direction? I've tried differentiating/integrating the series but this does not seem to be helpful.

You need to Use Dirchlet conditions at the end points

Dirichlet conditions - Wikipedia, the free encyclopedia

The Fourier series converges to the average of the right and left limit at that point so for your series you get
$\displaystyle f(\pi)=\frac{\lim_{x \to \pi^-}e^{x}+\lim_{x \to \pi^+}e^{x}}{2}=\frac{e^{\pi}+e^{-\pi}}{2}=\cosh(\pi)$
• Apr 4th 2011, 07:34 AM
StaryNight
Quote:

Originally Posted by TheEmptySet
You need to Use Dirchlet conditions at the end points

Dirichlet conditions - Wikipedia, the free encyclopedia

The Fourier series converges to the average of the right and left limit at that point so for your series you get
$\displaystyle f(\pi)=\frac{\lim_{x \to \pi^-}e^{x}+\lim_{x \to \pi^+}e^{x}}{2}=\frac{e^{\pi}+e^{-\pi}}{2}=\cosh(\pi)$

Many thanks, I was not aware of this theorem.