Thread: 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).



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.

Thanks in advance.

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).



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.

Thanks in advance.

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

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

Many thanks, I was not aware of this theorem.