Example 4.24. Investigate the convergence of the functional series

$$
\sum_{n=1}^{\infty} \frac{\sin n x}{e^{n x}}
$$
Solution. Apply the previous theorem:

$$
\varlimsup_{n \rightarrow \infty} \sqrt[n]{\frac{|\sin n x|}{e^{n x}}}=e^{-x} \varlimsup_{n \rightarrow \infty} \sqrt[n]{|\sin n x|}=e^{-x}
$$

From this, it is clear that for $x \in D=(0, +\infty)$, the series converges absolutely. For $x<0$, the series diverges. If $x=0$, the general term of the series is 0. The final result: the series converges for $x \geqslant 0$ and diverges for $x<0$.