Is there a close form solution for this series? I want to find close form solution for this series
$$\sum_{n=1}^{\infty}\cos\left(\frac{x}{3^n}\right)\sin^2\left(\frac{x}{3^n}\right)$$
Thanks for your help.
Hint

Set
$$f(x)=\sum\limits_{n=1}^{+\infty }{{{3}^{n-1}}{{\sin }^{3}}\left( \frac{x}{{{3}^{n}}} \right)}$$
we have
$$f'(x)=\sum\limits_{n=1}^{+\infty }{\cos \left( \frac{x}{{{3}^{n}}} \right){{\sin }^{2}}\left( \frac{x}{{{3}^{n}}} \right)}$$
Now use this identity and find $f(x)$
$${{\sin }^{3}}\theta =\frac{3}{4}\sin \theta -\frac{1}{4}\sin 3\theta $$
Playing around with the series numerically (plotting it etc) it seems it's equal to $\frac12 \sin^2(\frac{x}{2})$. I don't have a proof right now (and I'm too lazy to work one out), but it seems to work. Please feel free to check it yourself ;-)

With this in mind, you might be able to come up with something.

Cheers!