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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

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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.

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

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