Answer to Question #188118 in Real Analysis for Nikhil

Question

Test the series

∞

Σ(-1)^(n-1)× sinnx/n√n for absolute and

n=1 conditional convergence

Answer 1

Let us test the series $\sum_{n=1}^{\infty}\frac{(-1)^{n-1}\sin nx}{n\sqrt{n}}$ for absolute and conditional convergence.

Since for any $x\in\mathbb R$ we have that $|\frac{(-1)^{n-1}\sin nx}{n\sqrt{n}} |\le \frac{1}{n\sqrt{n}}=\frac{1}{n^{\frac{3}{2}}}$ and the s-series $\sum_{n=1}^{\infty}\frac{1}{n^{\frac{3}{2}}}$ is convergenent for $s=\frac{3}{2}>1$, we conclude that by comparison test the series $\sum_{n=1}^{\infty}\frac{(-1)^{n-1}\sin nx}{n\sqrt{n}}$ is absolute convergent for any $x\in\mathbb R$, and hence the series is also conditional convergent for any $x\in\mathbb R$.