Series is continuous by absolute convergence.

Question

I am reading some notes on analysis. At some point, the author defines : $F(x):=c_0 \frac{x^2}{2}+\sum_{n\neq 0} c_n \frac{e^{inx}}{(in)^2}$ and sais that by absolute convergence, F is a continuous function. I don't really understand what result this is and how it is applied here.

Answer 1

The absolute convergence of the series for $F(x)$ is equivalent to the convergence of $\sum_{n\neq 0}|c_n|/n^2$, which implies uniform convergence of the series, and since it consists of continuous functions, we're done.