Examine the series for convergence for summation of ((-1)^(n-1) sin(nx))/n^3
1) ∑n=1∞(−1)(n−1)⋅sin(nx)n3 if sin(nx)n3→ monotone for any x∈Nlimn→∞sin(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} 1) ∑n=1∞n3(−1)(n−1)⋅sin(nx) if n3sin(nx)→ monotone for any x∈Nlimn→∞n3sin(nx)=0 So series converges
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