Alternating cosine series, what is the closed form?

Question

What is the closed form for this series:

$$\sum _{n=0}^{\infty } \left(\frac{\cos \left(\frac{4 \pi}{2 n+1}\right)}{2 n+1}-\frac{\cos \left(\frac{4 \pi}{2 n+2}\right)}{2 n+2}\right)$$

if any?

I am looking for a simpler way to write it.

It converges to something and the decimal expansion starts: $0.4790882572765523426...$

Answer 1

We can write the given sum in this form

$$\sum_{n=1}^\infty(-1)^{n+1}\frac{\cos\left(\frac{4\pi }n\right)}{n}$$