Comparison Test for Series I am trying to prove that the series below converges by the comparison test. 
\begin{align}
\sum_{n=1}^\infty\frac{2n^3+5^n+5\log(n)}{13-n+8^n}
\end{align}
How do I show that:
\begin{align}
\frac{2n^3+5^n+5\log(n)}{13-n+8^n} \le 16\left(\frac{5}{8}\right)^n
\end{align}

Simple with asymptotic equivalence of functions:


*

*$2n^3+5^n+5\log n\sim_\infty 5^n$,

*$13-n+8^n\sim_\infty 8^n$,
so that


$$\frac{2n^3+5^n+5\log n}{13-n+8^n}\sim_\infty\frac{5^n}{8^n}=\Bigl(\frac 58\Bigr)^{\!n},\quad\text{a convergent geometric series.}$$
