saltydog

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.

[tex]\sum_{k=1}^{\infty}\frac{(-1)^{k+1}}k=ln(2)[/tex]

is shown by considering:

[tex]f(x)=ln(1+x)[/tex]

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

[tex]\sum_{k=1}^{\infty}\frac{1}{4k(2k-1)}[/tex]

is solved by considering the sum:

[tex]S=\frac{1}{2}[(1-1/2)+(1/3-1/4)+(1/5-1/6)+...[/tex]

and:

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

is solved by considering:

[tex]z=\sum_{n=0}^{\infty}w^n[/tex]

with:
[tex]w=e^{-a}[/tex]

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.