Evaluate the series $\sum_{n=1}^{\infty}\frac{n^3}{e^n \cdot n!}$

Asked Dec 31 '24 at 21:03

Active Dec 31 '24 at 21:51

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Evaluate $$\sum_{n=1}^{\infty}\frac{n^3}{e^n \cdot n!}$$

I could have solved the question if there wasn't that $e^n$ in the denominator by manipulation of the numerator and denominator of the fraction and shifting of the index.

But because of this $e^n$ I ain't got a clue about how to proceed. I also don't know any series that involves terms that occur in the expansion of the above function.

Any help is greatly appreciated.

edited Dec 31 '24 at 21:51

Gonçalo

15,869

asked Dec 31 '24 at 21:03

Blue Cat Blues

10,058

14

The $e$ isn't special, you want to try evaluating $\sum \frac{n^3 x^n}{n!}$ (then substitute $x = \frac{1}{e}$). Presumably you know what $\sum \frac{x^n}{n!}$ is. Can you figure out what $\sum \frac{nx^n}{n!}$ is? What about $\sum \frac{n^2 x^n}{n!}$? – Qiaochu Yuan Dec 31 '24 at 21:14
1

I think OP should have figured it out now. – Quý Nhân Dec 31 '24 at 21:30
@QiaochuYuan thanks...I solved it myself – Blue Cat Blues Jan 01 '25 at 03:56

Evaluate the series $\sum_{n=1}^{\infty}\frac{n^3}{e^n \cdot n!}$

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