Consider the following series:


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!


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.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.