# Calc6.46

Discuss the convergence or divergence of the series $\sum _{{n=1}}^{{\infty }}{\frac {n^{k}a^{n}}{n!}}$.
So here we have a power function of any order, k could be any number, 1, 100, 1,000,000, or even higher, and we also have an exponential function where $a$ can also be any number as high as you want. Yet, by the ratio test, we will show that a simple factorial function will eventually grow to be much greater than even the product of these two.
$\lim _{{n\rightarrow \infty }}\left|{\frac {{\frac {(n+1)^{k}a^{{n+1}}}{(n+1)!}}}{{\frac {n^{k}a^{n}}{n!}}}}\right|=\lim _{{n\rightarrow \infty }}\left|{\frac {(n+1)^{k}a}{n^{k}(n+1)}}\right|=0\,$