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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Sign up to join this communityDetermine 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
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.