# Math Help - convergent/divergent series

1. ## convergent/divergent series

test for convergence and/or divergence

$\sum_{n =1}^{\infty}n sin (1/n)$ is convergent?

i'm not sure what to do when it comes to trig functions in series.. integral test? comparison test?

2. Use the divergence test: If $\lim_{n \to \infty} a_n \neq 0$, then $\sum a_n$ diverges

Here, consider: $\lim_{n \to \infty} n \sin \tfrac{1}{n} = \lim_{n \to \infty} \frac{\sin \tfrac{1}{n}}{\tfrac{1}{n}}$

Let $\theta = \tfrac{1}{n}$. As $n \to \infty$, $\theta \to 0$.

So our limit becomes the familiar: $\lim_{\theta \to 0} \frac{\sin \theta}{\theta}$

3. Always first check whether the terms go to zero. If not, the series diverges. That's the situation here. There's no need for L'Hopital's Rule either. Let m=1/n...

$n\sin (1/n)={\sin m\over m}\to 1$ as $m\to 0$.