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Let $ f_n (x) = \left( \sin nx \right)/n^2. $ Show that the series $ \sum f_n(x) $ converges for all values of $ x $ but the series of derivatives $ \sum f'_n (x) $ diverges when $ x = 2n \pi, n $ an integer. For what values of $ x $ does the series $ \sum f''_n (x) $ converge?

This will comverge only when $\sin (n x)=0,$ otherwise $\sin (n x)$ will oscillate

between $-1$ and 1 as $n \rightarrow \infty .$

For it to be $0, n x$ has to be an integer multiple of $\pi,$ so

$x=n \pi, \quad$ where $n$ is an integer

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