Does the series converge or diverge? $\sum_{n=1}^\infty\frac{4^n+n}{n!}$

Question

Does the following series converge or diverge?

$$\sum_{n=1}^\infty\frac{4^n+n}{n!}$$

Some of the answers to your previous question from a few hours ago could be adapted to help here. For example, Cameron Buie's answer showed how you can see that $\sum \dfrac{2^n}{n!}$ converges, and the method can be adapted here, (e.g. using $n<4^n$ or the way Marvis did it). It would be appreciated by some (many, I think) if you indicate what you tried before asking. — Jonas Meyer, Jun 28 '12 at 01:14

score 9 · Accepted Answer · answered Jun 27 '12 at 23:46

9

$$\sum_{n=1}^{\infty} \dfrac{4^n + n}{n!} = \sum_{n=1}^{\infty} \dfrac{4^n}{n!} + \sum_{n=1}^{\infty} \dfrac{n}{n!} = \exp(4)-1 + \sum_{n=1}^{\infty} \dfrac1{(n-1)!} = \exp(4)-1 + \exp(1)$$

answered Jun 27 '12 at 23:46

I miss the easiest part. I didn't think about splitting it into two series! Thank you very much! – user1405177 Jun 27 '12 at 23:47

Does the series converge or diverge? $\sum_{n=1}^\infty\frac{4^n+n}{n!}$

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