As I was messing around in Maple (yes I'm a nerd), I noticed something.
It appears as if the infinite chain converges for all x.
It appears to be converging to a number very close to , but I don't think it actually does, unless it's really, really slowly.
The number (calculated for 7 sine-cosine pairs) is approximately .
Also, if you let the first term be cosine instead of sine, this chain gets close to .
Does anyone know anything about this?