I just wanted to check some of my solutions here concerning determining the convergence or divergence of a series using the divergence test. Can the divergence test be used to see if the following series diverges:
- $\sum_{n=1}^{\infty}(\frac{\pi}{3})^n$. The answer here is yes since the series diverges.
- $\sum_{n=1}^{\infty}(-1)^n \arctan(2n)$. The answer here is again, yes, since $\arctan(2n)$ converges to $\frac{\pi}{2}$, which means it diverges.
- $\sum_{n=1}^{\infty}\frac{n^2}{n^3+2}$. Sadly this one seems to not be applicable. It converges to 0, which means no information can be given.
- $\sum_{n=1}^{\infty}(\frac{n}{n+1})^n$. This would just be $(1)^n$ when $n$ approaches infinity. So it diverges.
Thank you for your time :D