Example of an infinite sum of functions $f_n(x)$ that converges to $x$, is there a typo in my book?

Question

I have a book that says the following:

Let $f_1(x), f_2(x), \dotsc$ a sequence of bounded functions with $f_1(x) + f_2(x) + \dotsb = x$, for example $$ f_1(x) = \frac{\sin x}{x}, \; f_n(x) = \frac{\sin xn}{xn} - \frac{\sin (n-1)x}{(n-1)x} $$

However, $$ \frac{\sin(x)}{x} + \sum_{n=2}^{\infty}\left( \frac{\sin(xn)}{xn}-\frac{\sin(x (n-1))}{x(n-1)} \right) = 0. $$ So is there an error or typo in the book and if so, what do you think the book meant to say? Or am I somehow misinterpreting what it says?

Answer 1

Provided $\sin x n \ne 0$ for any $n\in N$, we have $\sum_{n=1}^{\infty}f_n(x)=\lim_{m\to \infty}(\sin m x)/m x=0$ as you said. You are right. The book has an error,maybe a typo.