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Let $(c_n)$ be a decreasing sequence of positive numbers. If $\sum c_n \sin nx$ is uniformly convergent, then how to show that $\lim_{n\to\infty} n c_n = 0$?

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This isn't true in general. I think you want to assume you have uniform convergence on an interval of the form $I=[0,\delta]$. –  David Mitra Feb 7 '13 at 16:48
    
On what basis do you say that? You've not given any counter-examples. –  Saaqib Mahmuud Feb 15 '13 at 19:57
    
See this. So, for example, $\sum{1\over\sqrt n}\sin(nx)$ converges uniformly on $[\delta,2\pi-\delta]$, $\delta>0$ (but not on an interval containing $0$; hence my previous comment). I'm not sure how to prove your result, with the assumption of uniform convergence on an interval containing 0. –  David Mitra Feb 15 '13 at 20:25
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Hint: If $\sum u_n$ be a convergent series of positive real numbers and $\{u_n\}$ is a monotone decreasing sequence then $\lim_{n\rightarrow\infty} nu_n=0$

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Can you please also fill in the details and then also complete the proof of the original assertion? –  Saaqib Mahmuud Feb 7 '13 at 20:41
    
I'm still looking for a conclusive proof of my statement. Can anyone enlighten me on how to do it? –  Saaqib Mahmuud Feb 15 '13 at 19:58
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