Question #337547

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

1
Expert's answer
2022-05-09T12:27:45-0400

 1) n=1(1)(n1)sin(nx)n3 if sin(nx)n3 monotone  for any xNlimnsin(nx)n3=0 So series converges \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}




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