Answer to Question #287661 in Real Analysis for Reens

Question

Test the series: infinity sigma n=1 (-1)^ n-1 sin nx n sqrt n for absolute and conditional convergence

Answer 1

$\sum (-1)^ {n-1}\frac{ sin nx}{ n \sqrt {n}}$

$a_n=|(-1)^ {n-1}\frac{ sin nx}{ n \sqrt {n}}|=\frac{ |sin nx|}{ n \sqrt {n}}$

$b_n=\frac{1}{n\sqrt n}$

since $a_n\le b_n$ and $\sum b_n$ converges , $\sum a_n$ converges as well

so, series $\sum (-1)^ {n-1}\frac{ sin nx}{ n \sqrt {n}}$ converges absolutely