$$\sum_{r=0}^∞ \frac {\sin(r θ)}{3^r} $$
Given: $\sinθ=1/3$
I recently came across this question, and I tried numerous ways of solving it, but I could reach nowhere.
My initial approach was to expand uptil the first 3 terms to see if there was a pattern, but I don't think there is one. I also tried converting the question into a telescopic series, but I don't see how that can be done either. Any help would be appreciated!
Best Answer
The sum is $$\Im\sum_{r=0}^\infty\left(\frac13\exp i\theta\right)^r=\Im\frac{1}{1-\frac13\exp i\theta}=\Im\frac{1-\frac13\exp -i\theta}{\frac{10}{9}-\frac23\cos\theta}=\frac{3\sin\theta}{10-6\cos\theta}.$$For $\sin\theta=\frac13$, $\cos\theta=\pm\frac{2\sqrt{2}}{3}$, so the sum simplifies to $\frac{1}{10\mp 4\sqrt{2}}=\frac{5\pm 2\sqrt{2}}{34}$.