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This problem came from the Krantz text ($2^{nd}$ ed. ch. 9, prob. 17): Prove that the series $\displaystyle\sum_{j=1}^{\infty }{\frac{\sin{(jx)}}{j}}$ converges uniformly on compact intervals that do not contain odd multiples of $\displaystyle\frac{\pi}{2}$

Thank you in advance for your help

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Hmmm, are you sure is for odd multiples of $\frac{\pi}{2}$? This is the Fourier series for the sawtooth function, which is discontinuous at $x=0$, so this series cannot converge uniformly on any interval containing $0$. –  alejopelaez Aug 31 '11 at 14:03

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