Why does this function series converges uniformly?

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

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.