How to prove that the series $\sum_{n=1}^{\infty} (1+n!)/(1+n)!$ diverges?

Question

Wolfram alpha told me to use comparison test, so I am trying to compare it with the series $\sum_{n=1}^{\infty} n!/(1+n)!$. Am I on the right track? And if is the right way, how can I show that $\sum_{n=1}^{\infty} n!/(1+n)!$ diverges?

Answer 1

Yes you are doing great! Notice that your series is always greater than $\dfrac {n!}{(n+1)!}= \dfrac{n!}{n!(n+1)}=\dfrac {1}{n+1}$, which is basically the harmonic series which famously diverges

Answer 2

As Ovi answered, this is good work. After what Ovi wrote, you could even observe that $$u_n=\frac{n!+1}{(n+1)!}\implies u_n=\frac 1 {n+1}+\frac 1 {(n+1)!}$$