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Jun11-05, 09:59 PM
Sci Advisor
HW Helper
P: 1,593
Finding the sum of a series

It's a very interesting question. There are various techniques for solving such. For example,

[tex]\sum_{n=1}^{\infty} \frac{1}{n^2}=\frac{\pi^2}{6}[/tex]

This can be solved by using Fourier coefficients and Parseval's Theorem.


is shown by considering:


and differentiating and considering the Taylor series (thanks Daniel).


is solved by considering the sum:



[tex]\sum_{n=0}^{\infty} ne^{-an}[/tex]

is solved by considering:



and differentiating both the sum and the expression for the sum of the corresponding geometric series with respect to w.

Tons more I bet. Would be nice to have a compilation of the various methods for calculating infinite sums.