Sum of Series with Trigonometric Terms: Seeking Method to Find Exact Value

[rahl__]

Hi, I need to find the sum of such series:
$$\sum_{n=1}^{\infty}\sin{\frac{\pi}{2^n}}$$
i know that it's sum is less than $$\pi$$ but i don't know how to find the exact value.
thanks in advance for any help or clues

 

[NateTG]

What have you tried doing?

 

[rahl__]

first of all i'd like to correct myself as i don't really need to find that sum. i was just wondering whether my mathematical knowledge is big[?] enough to solve this problem, so what should I have really asked about is: what method would you choose to find that sum.

What have you tried doing?

I have used the comparative criterion (precisely this inequality: $$sin {x}\leq x$$)to find out that this series is convergent and that it's sum is equal or less than $$\pi$$, but i don't know what to do next. could you tell me what is the level of difficulty of this problem? is the solution rather complicated or can it be presented in a few lines? or which mathematical terms should i know in order to solve it on my own?