let $<a_n>$ be a sequence of real numbers such that $\sum|a_n-a_{n-1}|$ is convergent series. Show that power series $\sum a_n x^n$ converge on interval $(-1,1)$
How to approach . let $0<\alpha <1 $. Then i need to show series $\sum a_n \alpha^n$ is convergent. i am trying to show its sequence of partial sums $<S_n>$ is cauchy sequence . $$|S_n-S_m| \leq |a_{m+1}|+|a_{m+2}|+....+|a_n|$$ for $n\geq m$
But then how to use given convergent series? Any hint