[METHODS] Infinite series

For $r\ne1$ , first compute the finite partial sum

1+r+\cdots+r^N=\frac{1-r^{N+1}}{1-r}.

Then let $N\to\infty$ . The geometric series converges exactly when $|r|<1$ and in that case

\sum_{n=0}^{\infty}r^n=\frac1{1-r}.

If $|r|>1$ , the terms do not approach zero. If $r=1$ , the partial sums grow without bound, and if $r=-1$ , they oscillate.

If a term can be written as

a_n=b_n-b_{n+1},

then the finite partial sum is

\sum_{n=0}^{N}a_n =\sum_{n=0}^{N}(b_n-b_{n+1}) =b_0-b_{N+1}.

Take the limit of the single uncancelled tail to decide convergence and find the sum.

Cancelling inside an infinite expression, without first writing the finite sum, can hide a divergent tail.

Methods from Infinite series