Hey guys,
if |a_k|<|b_k| for large k, show that if (b_k)(x^k) converges on an open interval I, then (a_k)(x^k) also converges on I.
Please help.
Given a sequence the series converges if and only if where .
Therefore, converges if and only if where is chosen so large that .
Remember that an interval of converges of a series (centered as 0) is an interval centered at 0. If we look at the open interval then the series also converges absolutely. Now if then is on the open interval of convergence of and so converges too. But we have that and so converges too.