convergent/divergent series

**buttonbear** · March 27th 2009, 08:49 PM

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?

**o_O** · March 27th 2009, 10:55 PM

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: $\lim_{\theta \to 0} \frac{\sin \theta}{\theta}$

**matheagle** · March 27th 2009, 10:58 PM

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 $m\to 0$ .

Thread: convergent/divergent series

LinkBack

Thread Tools

Display

convergent/divergent series

Similar Math Help Forum Discussions

series convergent or divergent?

Show that the sum of a convergent and divergent series is divergent

Is this series convergent or divergent?

determine if series is absolutely convergent conditionally convergent or divergent

Series: Absolutely convergent, con convergent or divergent

Search Tags