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Check if the following series converge or diverge:$\sum_\limits{n=1}^{\infty}(1-\cos(\frac{\pi}{n}))$

I have tried the integral test since the series are decreasing to zero as $n\to\infty$, but $\int_1^\infty 1-\cos(\frac{\pi}{n}) dn=n-\sin(\frac{\pi}{n})|_0^\infty$, which diverges. I am not seeing what test could I applied that would deliver me the desired result convergence, since the book solution states that the series converge.

Question:

What do you think of the series? What test shall I use?

Thanks in advance!

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marked as duplicate by Robert Z calculus Jun 24 '18 at 14:03

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    $\begingroup$ Note that $\int 1 - \cos(\pi /x){\rm d}x$ is not $x - \sin(\pi/x)$. The integral actually converges. Also note that if this was correct then it would have proven that the series diverges. $\endgroup$ – Winther Jun 24 '18 at 14:00
  • $\begingroup$ @Winther What is the integral? How does it converge then? $\endgroup$ – Pedro Gomes Jun 24 '18 at 15:45
  • $\begingroup$ It's a non elementary function (try plugging it into WA to see it - to see its wrong just take the derivative of what you got and see you don't get the integrand back). It converges because the integrated behaves as $C/n^2$ for large $n $ just as the sum does and the integral of this converges. $\endgroup$ – Winther Jun 24 '18 at 15:48
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Hint: Use the fact that$$\lim_{n\to\infty}\frac{1-\cos\left(\frac\pi n\right)}{\frac1{n^2}}\in(0,+\infty).$$

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