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The sequence

$$\sum_{n=1}^{\infty} \frac{\sin (n)}{n}$$

converges to a specific value. Convergence tests of any kind are available in a quick search, but I'm struggling to find the proof that this sequence converges to $(\pi - 1)/2$. Here, complex logarithms are used.

I do not have any knowledge about series and sequences, just a real analysis / real functions background.

Is it possible to make the same proof, using only real functions? If yes, where can I find this proof?

BowPark
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    See https://math.stackexchange.com/questions/3361589/proof-that-sum-limits-n-1-infty-1n1-sinn-overn-1-over2 – lab bhattacharjee Sep 27 '19 at 07:57
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    Another one: https://math.stackexchange.com/questions/2031572/how-to-prove-sum-limits-n-1-infty-frac-sinnn-frac-pi-12-using-only. – Martin R Sep 27 '19 at 07:59
  • $f(x)=\text{sinc}(x)$ is an even entire function of order $1$. By the Euler-Maclaurin summation formula it follows that $$\sum_{n\geq 1}\text{sinc}(n)=-\frac{1}{2}+\int_{0}^{+\infty}\text{sinc}(x),dx=\frac{\pi-1}{2},$$ since the odd derivatives at the origin all equal zero. – Jack D'Aurizio Sep 27 '19 at 08:23

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