Calculus series question. Can anyone verify this answer?

Question

Question: Suppose you know that the series SUM(bn) from n=1 to infinity converges. What can you say about the series SUM(bn^n) from n=1 to infinity?
Answer: I believe the series converges absolutely by the ROOT test since limit |bn|=0 as n goes to infinity for a convergent series.

Answer 1

I believe that you are correct. I do have a degree in Math but havent used it in years so it is mostly gone. A function that converges to zero raised to a power will also converge to zero because 0 raised to a power is 0

Answer 2

Just to let you know
[Answer: I believe the series converges absolutely by the ROOT test since limit |bn|=0 as n goes to infinity for a convergent series.]

It is true that for a series to converge IbnI must tend to 0. However just because IbnI tends to 0 does not necessarily mean a series converges. For example the harmonic series (bn = 1/n) does not converge despite the fact 1/n tends to 0)

Answer 3

Your nomentclature is a bit fuzzy, but I suppose the term in the series is "b-sub n". Since each such term must be smaller than 1/n, raising each term to a positive power yields a smaller fraction. Thus, the series so created converges.