show this series converge uniformly to a continuous function

**silversand** · May 3rd 2009, 01:21 PM

$f(x)=\sum_{n\geq1}{a_{n}\cos(nx)}+\sum_{n\geq1}{b_ {n}\sin(nx)}\ \ \ \ \ \ with \ \ \ |a_{n}|\leq \frac{c}{n^{1+\epsilon}}, \ \ \ |a_{n}|\leq \frac{c}{n^{1+\epsilon}}$

**Opalg** · May 4th 2009, 12:07 AM

Originally Posted by silversand

$f(x)=\sum_{n\geq1}{a_{n}\cos(nx)}+\sum_{n\geq1}{b_ {n}\sin(nx)}\ \ \ \ \ \ with \ \ \ |a_{n}|\leq \frac{c}{n^{1+\epsilon}}, \ \ \ |a_{n}|\leq \frac{c}{n^{1+\epsilon}}$

Use the Weierstrass M-test for the uniform convergence; and the theorem that a uniform limit of continuous functions is continuous for the continuity of f(x).

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