I want to show that $\sum_{n=1}^{\infty}\frac{(-1)^n}{n+1}$ converges by using that theorem.
Theorem:
suppose
(a) the partial sum $A_n$ of $\sum a_n$ form a bounded sequence
(b) $\dots \le b_2 \le b_1 \le b_0$
(c) $\lim_{n\to\infty}b_n=0$
then $\sum a_n b_n$ converges.
Put $a_n=(-1)^n$ , $b_n=\frac{1}{n+1}$
I think other things are okay but not sure about $A_n$ is a bounded sequence.
Or I'm wrong from the beginning? This theorem is not suitable for this proof?