I want to show whether the following series converges pointwise? and whether it converges uniformly? $\sum_{n=1}^\infty \frac{arctan(nx)}{n^3} $ on $\mathbb{R}$.

For pointwise, I'm not sure how to approach it, can I have some hint.

For uniform convergence I used the M-test: $ |\frac{arctan(nx)}{n^3}| \leq \frac{0.5\pi}{n^3}$ and $\sum_{n=1}^\infty \frac{0.5\pi}{n^3} < \infty $. So it's uniformly convergent. Is this correct? If it does uniformly converge then it converges pointwise. Is there a way to check whether it converges pointwise without knowing it's uniformly convergent?

  • 2
    $\begingroup$ You actually proved the convergence is normal so it uniform so it is pointwise. That's correct. You proved the convergence is normal so it is absolute everywhere so it is pointwise (another way to check). $\endgroup$ – user371663 Feb 28 '18 at 15:14
  • $\begingroup$ Is the proof correct? $\endgroup$ – Jack Feb 28 '18 at 15:19
  • 1
    $\begingroup$ Yes it is correct. Just hope you realize you have proved normal convergence, not only the uniform one. $\endgroup$ – user371663 Feb 28 '18 at 15:23

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