For what value of $x$ the following series will converge $\sin x+2 \sin \frac x 3+4\sin \frac x 9+\ldots $? For what value of $x$ the following series will converge $\sin x+2 \sin \frac x 3+4\sin \frac x 9+8\sin \frac x {27}+\ldots $?
Work:
\begin{align}
\sin x+2 \sin \frac x 3+4\sin \frac x 9+8\sin \frac x {27}+\ldots & =\sum_{n=0}^\infty 2^n \sin \frac {x}{3^n}
\end{align}
Now, applying root test, we will get,
$$\limsup_{n\to \infty}\left| 2^n \sin \frac {x}{3^n} \right|^{\frac 1 n}=2\limsup_{n\to \infty}\left|\sin \frac {x}{3^n} \right|^{\frac 1 n} $$
After that, how to proceed to get the value of $x$ for which the series converges.

$$\lim_{n\to \infty} \left| \sin {x \over 3^n} \right|^{1/n} =
\lim_{n\to \infty} 
\left| \sin {x \over 3^n} \over {x \over 3^n}\right|^{1/n} \cdot \lim_{n\to \infty} \left| {x \over 3^n}\right|^{1/n}
= 1^0 \cdot \lim_{n\to \infty} \left| {x \over 3^n}\right|^{1/n}.
$$
You should be able to do the rest.
