Finding when Power Series Converge

We have seen how to get power series representations of functions using geometric series and using Taylor's theorem, both of which give ways to tell which x's give convergent series. Let us turn next to the problem of finding what values of x make a series converge where we are given the coefficients, but we are not told what the function being approximated is.

To start with let us note that if $\sum a_n$ and $\sum b_n$ converge, then so do $\sum (a_n+b_n)$ and $\sum (a_n-b_n)$ since their partial sums can be rearranged to be the sum and difference of partial sums of a_n and b_n.