My class was given a set of practice problems and I can't figure this one out.
If $\sum_{n=1}^{\infty}a_n3^n$ is convergent, then $\sum_{n=1}^{\infty}a_n(-2)^n$ is convergent.
My intuition wants to say this is true, but I can't figure out why. I can't apply the ratio test between the two because $a_n$ could be alternating, and limit comparison test between the first sum and $\sum_{n=1}^{\infty}a_n2^n$ doesn't work because the limit comparison test doesn't prove absolute convergence. I don't see any way to apply the alternating series theorem.
Essentially, this problem seems trivial if I know $a_n$ is either always positive or always negative, but any sort of alternation and it breaks down.