Thread: finding the sum of a series View Single Post
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,

$$\sum_{n=1}^{\infty} \frac{1}{n^2}=\frac{\pi^2}{6}$$

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

$$\sum_{k=1}^{\infty}\frac{(-1)^{k+1}}k=ln(2)$$

is shown by considering:

$$f(x)=ln(1+x)$$

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

$$\sum_{k=1}^{\infty}\frac{1}{4k(2k-1)}$$

is solved by considering the sum:

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

and:

$$\sum_{n=0}^{\infty} ne^{-an}$$

is solved by considering:

$$z=\sum_{n=0}^{\infty}w^n$$

with:
$$w=e^{-a}$$

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.