I need to find the partial sum formula for $\sum\limits_{n=0}^\infty \frac{n^3}{n!}$
I started by calculating some elements of the formula, but I could not find any possible patterns.
Could you please help me in finding the partial sum formula? I need it in order to be able to calculate the sum of the series. If there is any easier way to do this than finding a partial sum formula, please let me know.
Thank you in advance
Best Answer
We have that
$$\sum_{n=0}^\infty \frac{n^3}{n!}=\sum_{n=1}^\infty \frac{n^2}{(n-1)!}= \sum_{n=1}^\infty \frac{n^2-n+n-1}{(n-1)!}+\sum_{n=1}^\infty \frac{1}{(n-1)!}=$$
$$=\sum_{n=2}^\infty \frac{n-2+2}{(n-2)!}+\sum_{n=2}^\infty \frac{1}{(n-2)!}+\sum_{n=1}^\infty \frac{1}{(n-1)!}=$$
$$=\sum_{n=3}^\infty \frac{1}{(n-3)!}+3\sum_{n=2}^\infty \frac{1}{(n-2)!}+\sum_{n=1}^\infty \frac{1}{(n-1)!}=5e$$