Compute $\sum\limits_{n=1}^\infty \arctan\left(\frac{(-1)^{n+1}}{F_{n+1}(F_n+F_{n+2})}\right)$, where $F_n$ is $n$th Fibonacci number

Question

Compute $\sum\limits_{n=1}^\infty \arctan\left(\frac{(-1)^{n+1}}{F_{n+1}(F_n+F_{n+2})}\right)$, where $F_n$ is $n$th Fibonacci number.

Attempt:

I thought telescopic series method and this identity.

$$\arctan a+\arctan b=\arctan\left(\frac{a+b}{1-ab}\right)$$

But I am unable to find an analogy, a hint, or anything.

Answer 1

Hint.

Cassini's identity reads $$(-1)^n=F_n F_{n+2}-F_{n+1}^2$$

Using this, the general term inside your sum becomes $$u+v\over 1-uv$$

with $u=\frac{F_{n+2}}{F_{n+1}}$ and $v=-\frac{F_{n+1}}{F_{n}}$.