Need help to check alternating series criterion
$\frac{1}{n}-\frac{1}{n+1}=\frac{1}{n(n+1)}\gt\frac{1}{(2n+1)(n+1)}$. Therefore that extra term is taken care of.
$\frac{1}{n}-\frac{1}{n+1}=\frac{1}{n(n+1)}\gt\frac{1}{(2n+1)(n+1)}$. Therefore that extra term is taken care of.