$\sum\limits_{n=1}^{\infty}(-1)^{n+1}\frac{3^{n}}{3 \cdot 5 \cdot 7 \cdots (2n+1)}.$
I am trying to use Leibniz in order to prove that the series converges. I don't know if I am doing it correctly. Here it is.
I want to prove that it is decreasing.
$\frac{a_{n+1}}{a_{n}}= \frac{3}{2n+3}<1 ,$ so it is decreasing.
Then we want the $\lim\limits_{n \to \infty}a_{n}=0$. The problem is that the limit is not zero. So either I am missing something or the series does not converge.