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I want to show that $\sum^{\infty}_{n=1}\frac{\sin(nx)}n$ converges uniformly on $[a, 2\pi - a]$ for $0

Actually, I know $\sum^{\infty}_{n=1}\frac{\sin(nx)}n$ converges (by using Dirichlet test).

However, it is difficult for me to prove converge "uniformly".

How can I prove this?

Do I have to use Weierstrass M-test?

Then how?

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    Dirichlet's test can actually prove not only a mere pointwise convergence, but also uniform convergence!2012-09-10
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    And if you want to avoid a direct reference to this theorem, you may just take summation by parts and apply Weierstrass $M$-test.2012-09-10

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