evaluate series: sum [from n=0 to infty] 1/(2^n) tan(x/(2^n))

[newtagi]

Problem 76.3. Evaluate:

. . . . .\(\displaystyle \displaystyle \sum_{n\, =\, 0}^{\infty}\, \)\(\displaystyle \dfrac{1}{2^n}\, \tan\left(\dfrac{x}{2^n}\right)\)

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Hi, I am in trouble with that problem. This is my number theory project. I tried so much. And got only this.

\(\displaystyle \displaystyle \sum_{n\, =\, 0}^{\infty}\, \)\(\displaystyle \dfrac{1}{2^n}\, tg\, \left(\dfrac{x}{2^n}\right)\, =\, \dfrac{1}{2^m}\, ctg\, \left(\dfrac{x}{2^m}\right)\, -\, 2\, \cdot\, ctg\, (2x)\)[FONT=MathJax_Main]

Please need help with this problem. And I have limited time. Thanks in advance[/FONT]

 

[stapel]

Problem 76.3. Evaluate:

. . . . .\(\displaystyle \displaystyle \sum_{n\, =\, 0}^{\infty}\, \)\(\displaystyle \dfrac{1}{2^n}\, \tan\left(\dfrac{x}{2^n}\right)\)

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Hi, I am in trouble with that problem. This is my number theory project. I tried so much. And got only this.

\(\displaystyle \displaystyle \sum_{n\, =\, 0}^{\infty}\, \)\(\displaystyle \dfrac{1}{2^n}\, tg\, \left(\dfrac{x}{2^n}\right)\, =\, \dfrac{1}{2^m}\, ctg\, \left(\dfrac{x}{2^m}\right)\, -\, 2\, \cdot\, ctg\, (2x)\)[FONT=MathJax_Main]

Please need help with this problem. And I have limited time. Thanks in advance[/FONT]

How did you arrive at your result? Did you use reasoning similar to what is displayed here? What else are you needing? That is, for what, specifically, are you asking?

Thank you!