showing that an infinite series converge

Question

im trying to prove that the infinite seires :

converges. ive managed to show that it does not absloutly converge. i cant use Dirichlet's test and i cant use Abel's test since that series dosent meet the demands of the theorems. any ideas? thanks.

Answer 1

$|\frac {3^{n}+4^{n}} {2^{n}+(-1)^{n} 5^{n}}| \leq \frac {(4.5)^{n}+(4.5)^{n}} {5^{n}-2^{n}} <\frac {2 ((4.5)^{n})} {5^{n}-\frac 1 25^{n}}$. Can you finish?

Answer 2

The series (with terms $a_n$) is absolutely convergent: $$ |a_n|\le \frac{2\cdot 4^n}{5^n-2^n}\le (4/5)^n $$