Prove or disapprove $\pi = 3 + \sum_{n=1}^{\infty} \frac{(-1)^{n}(x_1^n + x_2^n)}{n+2}$,
where $x_1$ and $x_2$ are the roots of $2x^2+2x+1=0$.
I've tried rewriting the series but it didn't work. There is definitely a trick but I don't see it. I will appreciate any help. Thanks!
Borrowing @MatthewPilling's calculation, the series (not including the $3$) is $\sum_\pm\sum_{n\ge1}\frac{z_\pm^n}{n+2}$ with $z_\pm:=\tfrac{1}{\sqrt{2}}\exp\frac{\pm\pi i}{4}$, i.e.$$-\sum_\pm\left(\tfrac{z_\pm+\ln(1-z_\pm)}{z_\pm^2}+\tfrac12\right)=-\sum_\pm\left(\sqrt{2}\exp\tfrac{\mp\pi i}{4}-\tfrac{\pi}{2}\pm i\ln2+\tfrac12\right)=\pi-3.$$Numerical analysis agrees.