Consider the series $f(x) = \sum_{n=1}^{\infty} a_n \sin nx$, where \begin{align} a_n = \begin{cases} \frac{4}{n \pi}, &n \text{ odd} \\ 0, &n \text{ even} \end{cases}. \end{align} It is easily seen that \begin{align} f(x) = \begin{cases} -1, & - \pi < x < 0 \\ 0, & x= 0 \\ 1, & 0 < x < \pi \end{cases}. \end{align}

Does this series converge uniformly?


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No, because a series of continuous functions cannot converge uniformly to a discontinuous function.


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