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

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

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