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.
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 $x<1$.