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?