Prove that the series $\displaystyle \sum_{n=1}^{\infty}\frac{\cos nx}{\ln (n+1)}$ is not a Fourier series of a $2\pi$ periodical Riemann integrable function.
Solution:
It follows immediately from Parseval's identity, since:
$$\sum_{n=1}^{\infty}a_n^2 =\frac{1}{\pi}\int_{-\pi}^{\pi}f^2(x)\,{\rm d}x \Leftrightarrow \sum_{n=1}^{\infty}\frac{1}{\ln^2 (n+1)}=\frac{1}{\pi}\int_{-\pi}^{\pi}f^2(x)\,{\rm d}x$$
The LHS clearly diverges to $+\infty$ due to Pringsheim test, since $n/ \ln^2 (n+1)\rightarrow +\infty$. The result follows.
Solution:
It follows immediately from Parseval's identity, since:
$$\sum_{n=1}^{\infty}a_n^2 =\frac{1}{\pi}\int_{-\pi}^{\pi}f^2(x)\,{\rm d}x \Leftrightarrow \sum_{n=1}^{\infty}\frac{1}{\ln^2 (n+1)}=\frac{1}{\pi}\int_{-\pi}^{\pi}f^2(x)\,{\rm d}x$$
The LHS clearly diverges to $+\infty$ due to Pringsheim test, since $n/ \ln^2 (n+1)\rightarrow +\infty$. The result follows.
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