I am new to calculus, and I just wanted to check my reasoning re the following:
Given is the following series:
$$\sum n^s·e^{-n}, s \ge0$$
I was asked to prove that this series converges.
My reasoning was:
$$\lim_{n\to \infty} \frac{(n+1)^s}{e^{(n+1)}}·\frac{e^n}{n^s}=\lim_{n\to \infty} \frac{(n+1)^s}{en^s}=\frac {1}{e}$$
As $\frac {1}{e}<1$, the series converges.
Is this correct?
Thank you!
Your proof is definitely valid as an application of the ratio test.
One minor nitpick: it would've been more appropriate to indicate the index of summation, i.e.
$$\sum_{n=0}^\infty \;\;\; \text{instead of} \;\;\; \sum$$
While it's okay to use the latter when the indices or the space over which you're summing is well understood, it's not necessarily true here, in my opinion. But this may well be just a personal issue since I struggled for a sec in figuring out which variable you were summing over.
(For completeness' sake I assume it was something like the former, an infinite summation starting at some finite index, i.e. $0, 1,$ etc., and was a summation over $n$.)