# Absolutely convergent vs conditionally convergent?

Why is the series $a_n={1\over2^n+n}$ convergent whereas the series $a_n={3^n+1\over4^n+5}$ is absolutely convergent? Since both use the comparison test to the geometric series, what makes one abs. convergent and the other not? I know a series converges absolutely if $|a_n|$ converges so why the difference?