Q:

Confused as to which test to use to determine if series converges or diverges

For the question $$\sum^\infty_{n=0}\frac{n^2}{2^n+1}$$I first tried the root test as the denominator was a number to the power of n, but it would result in the numerator having a power to the n so I scrapped that idea. I tried to use the divergence test and then l'hopital's rule as both numerator and denominator went to infinity but I feel like that's over-complicating the question and that there's an easier test for it. Please help.

A:

You can combine the comparison test $\sum \frac{n^2}{2^n+1} < \sum \frac{n^2}{2^n}$ with the ratio test, since
$$
\lim_{n\to\infty} \bigg| \frac{(n+1)^2/2^{n+1}}{n^2/2^n} \bigg| = \lim_{n\to\infty} \frac{n^2+2n+1}{2n^2} = \frac12,
$$
indicating convergence.
You can also use the ratio test directly on the original sum; the limit to evaluate is a bit more complicated, but still very doable.

