Sequence of series

Let $f_n: \mathbb{R} \rightarrow \mathbb{R}$ be defined as:

$$f_n(x)=\frac{\cos nx}{n^2}, \; \; n \in \mathbb{N}$$

Prove that the series $\displaystyle \sum_{n=1}^{\infty} f_n$ converges uniformly to a function $f:\mathbb{R} \rightarrow \mathbb{R}$ and that:

$$\int_0^{\pi/2} f(t)\, {\rm d}t = \sum_{k=0}^{\infty} \frac{(-1)^k}{(2k+1)^3}$$

Solution



That the series converges uniformly is an immediate consequence of the Weierstrass M test, since:

$$\left | \sum_{n=1}^{\infty}\frac{\cos nx}{n^2} \right |\leq \sum_{n=1}^{\infty}\frac{1}{n^2}$$

For the second part we have that:

$$\begin{aligned}
\int_{0}^{\pi/2}f(t)\, {\rm d}t &=\int_{0}^{\pi/2} \sum_{n=1}^{\infty} \frac{\cos nx}{n^2}\, {\rm d}x \\
 &\overset{(*)}{=}\sum_{n=1}^{\infty}\frac{1}{n^2}\int_{0}^{\pi/2}\cos nx \,{\rm d}x \\
 &= \sum_{n=1}^{\infty}\frac{\sin \frac{\pi n}{2}}{n^3} \\
 &= \sum_{k=0}^{\infty}\frac{(-1)^k}{\left ( 2k+1 \right )^3}
\end{aligned}$$

since $\sin \frac{\pi n}{2}$ introduces the alternating term .

$(*)$ This is justified due to uniform convergence.