test for convergence and/or divergence
$\displaystyle \sum_{n =1}^{\infty}n sin (1/n)$ is convergent?
i'm not sure what to do when it comes to trig functions in series.. integral test? comparison test?
Use the divergence test: If $\displaystyle \lim_{n \to \infty} a_n \neq 0$, then $\displaystyle \sum a_n$ diverges
Here, consider: $\displaystyle \lim_{n \to \infty} n \sin \tfrac{1}{n} = \lim_{n \to \infty} \frac{\sin \tfrac{1}{n}}{\tfrac{1}{n}}$
Let $\displaystyle \theta = \tfrac{1}{n}$. As $\displaystyle n \to \infty$, $\displaystyle \theta \to 0$.
So our limit becomes the familiar: $\displaystyle \lim_{\theta \to 0} \frac{\sin \theta}{\theta}$