# Finding another series for a given series.

For each sequence $$a_n$$ find a number r such that $$\frac{a_n}{r^n}$$ has a finite non-zero limit. (This is of use, because by the limit comparison test the series $$\displaystyle \sum_{n=1}^\infty a_n$$ and $$\displaystyle \sum_{n=1}^\infty r^n$$both converge or both diverge.) A. $$a_n = (5 + 5 ^ n)^{- 4}$$ r =

This is impossible. There are many sequences $a_n$ for which there is no such a $r$.
Moreover, your assertion that ${a_n}$ and $r^{n}$ converge or not simultaneously is only true if the sequence ${a_n}$ has positive terms.
On the other hand, $\sum (5 + 5^n)^{-4}$ converges because $0\leq (5 + 5^n)^{-4} \leq (5^{-4})^{n}$.
• Can you please show your steps of how you came to the conclusion of $$0\leq (5 + 5^n)^{-4} \leq (5^{-4})^{n}$$ @RobertFuster – M.E. Apr 15 '14 at 9:03