$f_n (x):= \dfrac{\sin (2n\pi x)}{2n\pi}.$

I am working with a function series: $\sum_{n=1}^{\infty} f_n(x).$

My mathematics book says that this function series converges uniformly in every closed interval that contains no integer.

But I cannot understand why:

$$\bigg| \sum_{n=1}^{\infty} f_n(x) \bigg| \leqq \sum_{n=1}^{\infty} \dfrac{1}{2n\pi}$$ but the right side of this inequality doesn't converge.

Furthermore, I don't understand why my book says "in every closed interval that contains no integer."

Even if $x$ is integer, or $x\in \mathbb{Z} $, $\quad \sum_{n=1}^{\infty} f_n(x)=0, \quad$ so I think this series converges uniformly.


Accorging to Theorem 7.2.2, part (1), page 112 of Fourier Series by Edwards this statement is false.


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