Answer to Question #337547 in Calculus for surya

Question

Examine the series for convergence for summation of ((-1)^(n-1) sin(nx))/n^3

Answer 1

$\begin{array}{l} \text { 1) } \sum_{n=1}^{\infty} \frac{(-1)^{(n-1)} \cdot \sin (n x)}{n^{3}} \quad \text { if } \frac{\sin (n x)}{n^{3}} \rightarrow \text { monotone } \\ \text { for any } x \in N \quad \lim _{n \rightarrow \infty} \frac{\sin (n x)}{n^{3}}=0 \\ \text { So series converges }\\ \end{array}$