# Is the series $\sum_{n=1}^{\infty}\frac{(1+\frac{n}{\mathrm{e}})^n}{n\mathrm{!}}$ convergent?

Test the convergence of the series
$$\sum_{n=1}^{\infty}\frac{(1+\frac{n}{\mathrm{e}})^n}{n\mathrm{!}}$$
Thanks!

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What tests have you tried? What can you say about the numerator? –  Calvin Lin Jan 29 '13 at 7:53
I have tried cauchy root test, it gives no conclusion. –  asimath Jan 29 '13 at 7:55
You will find the Stirling approximation to the factorial works very nicely. One could get away with somewhat less. –  André Nicolas Jan 29 '13 at 7:58

Observe that by Stirling's approximation we always have $$n!\le e\sqrt{n}\left(\frac{n}{e}\right)^n$$ and so we see that $$\sum_{n=1}^\infty\frac{\left(1+\frac{n}{e}\right)^n}{n!}\ge \sum_{n=1}^\infty\frac{\left(\frac{n}{e}\right)^n}{e\sqrt{n}\left(\frac{n}{e}\right)^n}=\frac{1}{e}\sum_{n=1}^\infty\frac{1}{\sqrt{n}}=\infty$$ thus the series diverges.
we have $$\left(\frac{n}{\mathrm{e}}\right)\leq \left(1+\frac{n}{\mathrm{e}}\right)$$
And so we have$$\frac{{\left(\frac{n}{\mathrm{e}}\right)}^n}{n\mathrm !}\leq \frac{{\left(1+\frac{n}{\mathrm{e}}\right)}^n}{n\mathrm !}$$
Now the series $\sum_{n=1}^{\infty}\frac{{\left(\frac{n}{\mathrm{e}}\right)}^n}{n\mathrm !}$ is divergent by logarithmic test. So the given series is divergent.