Two simple series

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}

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Nobody knows how to calculate those two series. But some of us know how to decide whether or not they converge. Will that do? –  Gerry Myerson Feb 4 '13 at 12:36
@Ryan, this is your 31st question in the last 40 days, and in none (well, let us write "almost none" since I didn't check them all) of them you give some original thought, show some self effort and work, some insight...are you trying to have others to do your homework? –  DonAntonio Feb 4 '13 at 12:40
The title also is annoying: simple series...if they are simple then why don't you do the work alone? –  DonAntonio Feb 4 '13 at 12:41
@GerryMyerson : Yes, that would help :) If there is no exact value –  Ryan Feb 4 '13 at 12:43
@DonAntonio : Because I think the second one may done with some trig identity . –  Ryan Feb 4 '13 at 12:46

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.

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Thx! Both converges –  Ryan Feb 4 '13 at 13:08

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 .

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

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

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