TITLE: Convergence of power series under a given condition
QUESTION [0 upvotes]: The question i am trying to figure out is:

If the series $\sum_{1}^\infty |a_n-a_{n-1}|$ is convergent, then
  where should the power series $\sum_{0}^\infty a_n{x^n} $ converge?

The answer given simply says it would converge on SOME set that contains the open interval (-1,1). I didn't get the concept/ theory that works here. Any helpful reference or supporting proof would be appreciated.

REPLY [1 votes]: If $\sum_{n=1}^\infty|a_n-a_{n-1}|$ converges, so does $\sum_{n=1}^\infty(a_n-a_{n-1})$. This implies that
$$
a_n=a_0+\sum_{k=1}^n(a_k-a_{k-1})
$$
converges, and in particular, it is bounded. From this, it follows easily that $\sum_{n=0}^\infty a_nx^n$ converges absolutely if $x<1$.
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