Find an appropriate power series and apply the ratio test to show the following infinite sum converges: $$ \sum_{n=0}^\infty \frac{n^2}{5^n} $$
I'd simply do it like this: $$ \text{Let } a_n = \frac{n^2}{5^n}. \text{Then by the ratio test we have}\\ \begin{align*} \left|{\frac{a_{n+1}}{a_n}}\right|= \left|\frac{(n+1)^2}{5^{n+1}}\cdot \frac{5^n}{n^2}\right| = \left|\frac{n^2+2n+1}{5n^2}\right| \end{align*}\\ \implies \lim_{n\to\infty} \frac{n^2+2n+1}{5n^2} = \lim_{n\to\infty} \frac{1+\frac{2}{n}+\frac{1}{n^2}}{5} = \frac{1}{5} < 1 $$
I'm confused since I thought I could directly apply the ratio test on this series. (without finding a power series)
I guess my professor wants me to solve it like that, but I have no idea what kind of convergent power series I could (upper-)bound this with.