Methods from Power series

For a power series centred at $a$:

1. apply the ratio or root test when its limit exists and is decisive; otherwise estimate the coefficients by another convergence test;
2. rewrite that condition as $|x-a|<R$;
3. record the open interval $(a-R,a+R)$;
4. if $0<R<\infty$, substitute $x=a-R$ and $x=a+R$ into the original series;
5. apply the appropriate ordinary series test at each finite endpoint.