Calculating the summation $\sum_{n=0}^{\infty} \frac{n^3}{3^n}$

Asked Mar 24 '20 at 23:13

Active Mar 24 '20 at 23:13

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Evaluate $$\sum_{n=0}^{\infty} \frac{n^3}{3^n}.$$

I started my using casework, listing out some terms when plugging values in. $0, \frac{1}{3}, \frac{8}{9}, 1.$ I noticed that there is no common ratio involved in these terms. I'm not sure if I can do any partial fraction decomposition here either. I'm not sure if it telescopes.

asked Mar 24 '20 at 23:13

Frost Bite

Here there is partial sum: perhaps to prove it you can use induction, but I think that it is difficult since the partial sum has transcendental form. – Antonio Maria Di Mauro Mar 24 '20 at 23:20
Thanks, I think these helps after reading both of them. – Frost Bite Mar 24 '20 at 23:22
Anyway I repeat that the induction is too much complicate: the partial sum has a trascendental form! – Antonio Maria Di Mauro Mar 24 '20 at 23:23
@AntonioMariaDiMauro I was writing up an answer that does the summation without calculus. It is doable by writing as a multi sum and then switching the order of summation. – Ninad Munshi Mar 24 '20 at 23:24
@NinadMunshi Okay, so we are waiting for your answer. – Antonio Maria Di Mauro Mar 24 '20 at 23:25
@AntonioMariaDiMauro lol! The post is closed, it will not accept answers anymore, hence why I wrote a comment. – Ninad Munshi Mar 24 '20 at 23:27
How come it is closed? – Frost Bite Mar 24 '20 at 23:27
Also see https://math.stackexchange.com/a/3525783/442 – GEdgar Mar 24 '20 at 23:35

Calculating the summation $\sum_{n=0}^{\infty} \frac{n^3}{3^n}$

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