# Determine whether the series $\sum\limits_{n=1}^\infty \frac{π^n+n}{3^n+n!}$ is convergent or divergent.

Determine whether the series

$\sum\limits_{n=1}^\infty \frac{π^n+n}{3^n+n!}$

is convergent or divergent.

Is a positive series and I can see that for $x\to \infty$

$\frac{π^n}{n!}$

and I don't know how to use the comparison test

## 1 Answer

It converges. $$0\le \frac{\pi^n + n}{3^n+n!}\le \frac{2\pi ^n}{n!}$$ and $$\lim_{n\to\infty}\frac{\frac{2\pi^{n+1}}{(n+1)!}}{\frac{2\pi^n}{n!}}=0,$$ so $\sum_{n=1}^{\infty}\frac{2\pi^n}{n!}$ converges by ratio test. Therefore, given series converges by the comparison test.

• you even can compute $\sum_{n=1}^\infty 2\frac{\pi^n}{n!} = 2 (e^{\pi}-1)$ – MJ73550 Apr 22 '16 at 11:04
• @MJ73550 Thanks for giving exact value of that. It is from $e^x =\sum_{n=0}^{\infty} \frac{x^n}{n!}$, right? – choco_addicted Apr 22 '16 at 11:06
• @choco_addicted Yep it is. – Eff Apr 22 '16 at 11:07
• @choco_addicted Why the numeratior is $2\pi^n$ ? thanks – GiovanS Apr 22 '16 at 11:09
• I replaced $n$ with $\pi^n$, because $n < \pi^n$ is true for $n\in \mathbb{N}$. – choco_addicted Apr 22 '16 at 11:11