Two simple series

Question

I dont know how to calculate these two series: $$\begin{align} & \sum\limits_{n=1}^{\infty }{\frac{n+3}{{{n}^{3}}+\ln n}} \\ & \sum\limits_{n=1}^{\infty }{\left( 1-\cos \frac{\pi }{n} \right)} \\ \end{align}$$

Answer 1

First one --- do a comparison to $\sum n^{-2}$.

Second one --- see whether you can find useful bounds on $\cos(\pi/n)$.

That is all --- you're on your own.

Answer 2

for 2>:

$$ 1 - \cos\frac{\pi}{n} = 2\sin^2\frac{\pi}{2n} <\frac{\pi}{2n^2}$$ then use convergence test to show series is convergent .

Answer 3

For the second sum, we can have it in another form

$$\sum\limits_{n=1}^{\infty }{\left( 1-\cos \frac{\pi }{n} \right)} =\sum _{n=1}^{\infty }{\frac { \left( -1 \right) ^{n+1}{\pi }^{2\,n} \zeta\left( 2\,n \right) }{ \left( 2\,n \right) !}} \sim 4.870718962.$$