Calculate $\frac{\pi}{2}-\sin(\frac{\pi}{2})+\sin(\sin(\frac{\pi}{2}))-\sin(\sin(\sin(\frac{\pi}{2})))\cdots$

Question

I would like to calculate following sum: $$\frac{\pi}{2}-\sin\left(\frac{\pi}{2}\right)+\sin\left(\sin\left(\frac{\pi}{2}\right)\right)-\sin\left(\sin\left(\sin\left(\frac{\pi}{2}\right)\right)\right)+\cdots\approx1.02$$ It can be rigorously defined as follows: $$f_n=\sin(f_{n-1})$$ $$f_0=\frac{\pi}{2}$$ $$\sum_{n=0}^\infty f_n(-1)^n$$ The sum converges rather slowly. Approximating with $5 \cdot 10^6$ terms gives result $1.020\pm0.0002$
Thanks for all the help