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

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SUMMARY

The discussion centers on finding the exact value of the series \(\sum_{n=1}^{\infty}\sin{\frac{\pi}{2^n}}\). The user has established that the series converges using the comparative criterion, specifically the inequality \(\sin{x} \leq x\), and knows that the sum is less than \(\pi\). However, they seek guidance on the appropriate methods and mathematical concepts required to derive the exact value of the series.

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Hi, I need to find the sum of such series:
[tex]\sum_{n=1}^{\infty}\sin{\frac{\pi}{2^n}}[/tex]
i know that it's sum is less than [tex]\pi[/tex] but i don't know how to find the exact value.
thanks in advance for any help or clues
 
Last edited:
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What have you tried doing?
 
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: [tex]sin {x}\leq x[/tex])to find out that this series is convergent and that it's sum is equal or less than [tex]\pi[/tex], 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?
 
Last edited:

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