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 $\displaystyle \{ x_n\}$ the series $\displaystyle \sum_{n=1}^{\infty}x_n$ converges if and only if $\displaystyle \sum_{n=N}^{\infty} x_n$ where $\displaystyle N>1$.
Therefore, $\displaystyle \sum_{k=1}^{\infty}a_kx^k$ converges if and only if $\displaystyle \sum_{k=K}^{\infty}a_kx^k$ where $\displaystyle K>1$ is chosen so large that $\displaystyle |a_k| < |b_k|$.
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 $\displaystyle x\in I$ then $\displaystyle x$ is on the open interval of convergence of $\displaystyle \sum_{k=K}^{\infty}b_kx^k$ and so $\displaystyle \sum_{k=K}^{\infty}|b_kx^k|$ converges too. But we have that $\displaystyle |a_kx^k| \leq |b_kx^k|$ and so $\displaystyle \sum_{k=K}^{\infty}a_kx^k$ converges too.