Does sequences related to function for $lcm(1,2,3 \cdots n)$ exists?

Question

This just came out of curiosity let $$L(n)=lcm(1,2,3 \cdots n)$$ and I know that we can write this with the help of some product involving primes and all . But what I am interested is in

Does $$\sum^{\infty}_{n=1} \frac{L(n)}{(n!)^2}$$ exists ? If yes than what is it equal to ?And another thing is does $$\sum_{n=1}^{\infty} \frac{1}{L(n)}$$ exists ? If yes is there a close form for it?

Answer 1

$$0\leq \Sigma \frac{l(n)}{(n!)^2}\leq\Sigma \frac{n!}{(n!)^2}=\Sigma \frac{1}{n!}=e-1 $$