How to prove that $\sum_{n=1}^{\infty}\frac{1!+2!+\cdots+n!}{(2n)!}$ converges?
Solution 1:
Since $(2n)!=(n!)^2\dbinom{2n}{n}>(n!)^2$, then $$\sum_{n=1}^{\infty}\frac{1!+\dots+n!}{(2n)!}<\sum_{n=1}^{\infty}\frac{n\cdot n!}{(n!)^2} =\sum_{n=1}^{\infty}\frac{1}{(n-1)!}=e<\infty.$$