How would you prove the series $\{\frac{1}{n^2+1}\}$ from 0 to infinity converges?

Question

I understand you would usually compare it with $\{\frac{1}{n^2}\}$ if $n=1$, which converges, but this series starts at $n = 0$ which is undefined for $\{\frac{1}{n^2}\}$ which is the start point of this series.

Answer 1

$$\sum_{n=0}^\infty \frac 1{n^2+1}=1+\sum_{n=1}^\infty \frac 1{n^2+1}$$ Adding one (or any finite number) cannot change whether the sum converges, so the comparison you mention with $\sum \frac 1{n^2}$ is sufficient to establish convergence.