Sum k=1...infinity (k^90*e^-k) converges or diverges?

Mar 2016
5
0
Denver, CO
Hi, all. I got this one incorrect and I'm wondering how to get it started. Any thoughts on a good first step? Thanks!

Converges or diverges?
Sum k=1...infinity (k^90*e^-k)
 

skeeter

MHF Helper
Jun 2008
16,217
6,765
North Texas
$\displaystyle \sum_{k=1}^\infty \dfrac{k^{90}}{e^k}$

root test ...

$\displaystyle \lim_{k \to \infty} \left(\dfrac{k^{90}}{e^k}\right)^{1/k}$

$\displaystyle \lim_{k \to \infty} \dfrac{(k^{1/k})^{90}}{e} = \dfrac{1}{e} < 1$

therefore, converges.

I believe that the ratio test may also be used to show convergence.