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.
[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(2k1)}[/tex]
is solved by considering the sum:
[tex]S=\frac{1}{2}[(11/2)+(1/31/4)+(1/51/6)+...[/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.
