Your problem requires cleverness i do not have at this time so i am looking forward to a solution posted by another MHF member. The series is interesting.
WIA is also stumped.
series(sin(nx))/(2^n&# 41; , n = 1 to infinity) - Wolfram|Alpha
I'll replace with for convenience.
Using the identity , your series may be rewritten as
Individually, these two sums are convergent geometric series (do you know why?). Calculate the sum for each term and add them up. I have verified that you do indeed get the desired result (after some algebraic manipulations), but I have no desire to type it out. You should be able to work it out yourself.
You may need to justify why you are able to break the sum into two infinite series, but this is a straightforward application of limit theorems since both your series converge.