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Investigate the convergence of the series on the interval $x \in [0;1]$

$$\sum_{n = 1}^{\infty} \frac{x}{1 + n^2x^2} $$

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Hint: $$\sum_{n = 1}^{\infty} \frac{x}{1 + n^2x^2}=\frac{\pi}{2}(\coth \frac{\pi}{x}-\frac{x}{\pi})$$

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    $\begingroup$ Are you sure? The left-hand side is defined in $0$, while the right-hand side is not. $\endgroup$ – Alex M. Oct 30 '16 at 19:34
  • $\begingroup$ $\lim_{x\rightarrow 0^{+}}\coth \frac{\pi}{x}=1$ $\endgroup$ – E.H.E Oct 30 '16 at 19:38
  • $\begingroup$ @E.H.E: ...in which case for $x=0$ the left-hand side is $0$ and the right-hand side is $\frac \pi 2$. Maybe that equality is valid on $(0, 1]$, I don't know, but it surely is not valid on $[0,1]$. $\endgroup$ – Alex M. Oct 30 '16 at 19:40
  • $\begingroup$ @Winther: Even interpreted as a limit, the equality is not true, see my comment above. $\endgroup$ – Alex M. Oct 30 '16 at 19:45
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    $\begingroup$ @AlexM. Yes you are quite right. The series does not converge uniformly on any interval containing $x=0$ so the limit is not guaranteeds to equal the right hand side function value there. I should have read the question I linked to better. Thanks for correcting me. $\endgroup$ – Winther Oct 30 '16 at 19:48

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