How do you determine the nature of this series

Consider the following series:

$\sum{(-1)^n\sin{(\pi\sqrt{1+n^2})}}$

We want to determine if the series diverges or not.

I can prove that all the terms of the series are positive, but that's all. I have no clue how to prove that the series converges (or diverges?). Also, I thought of something like: $\sqrt{1+n^2}\sim n$ and so the series converges, but I doubt it is mathematically correct.

Thank you!

1 Answer

Well, you need to look at $\sqrt{1+n^2}$ as $n\sqrt{1+{1\over n^2}}=n+{1\over2n}+o({1\over n})$. Then indeed, we see that all terms are positive, and that they tend to zero, but too slowly (about as $1\over n$). All in all, your series diverges for the same reason as harmonic series.

The $\sqrt{1+n^2}\sim n$ part, though true, is quite irrelevant to the convergence.