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