Prove that the series

$$\sum_{k=2}^∞ \sin (kx)/k\ln^2(k)$$

is absolutely and uniformly convergent on $\mathbb{R}$. If the sum of the series is denoted by $f(x)$ prove that $f$ is differentiable at each point of ($0; 2\pi$), and find $f'$.

I already proved that it absolutely and uniformly converges, but I don't know how to prove that it is differentiable. Any help please?

  • 3
    $\begingroup$ Integration term-by-term is easier to justify than differentiation. So consider the term-by-term derivative of your series, and investigate whether it can be integrated. $\endgroup$ – GEdgar May 9 at 12:58

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