敛散性问题

castelu

幂级数在|x|<r收敛，则在|x|<r一致收敛，所以逐项可积
$
\int_0^1{\frac{{\ln(1-x)}}{x}dx}\\
=\int_0^1{\sum\limits_{n=1}^\infty{\frac{{{{(-1)}^{n-1}}}}{{nx}}{{(-x)}^n}}dx}\\
=\sum\limits_{n=1}^\infty{\int_0^1{\frac{{{{(-1)}^{2n-1}}}}{n}{x^{n-1}}dx}}\\
=-\sum\limits_{n=1}^\infty{\frac{1}{{{n^2}}}}=-\frac{{{\pi^2}}}{6}
$