a) Use mathematical induction to prove that $$(1)(1!)+(2)(2!)+(3)(3!)+...+(n)(n!)=(n+1)!-1,$$ where n $\in \Bbb Z^+$.
b) Find the minimum number of terms of the series for the sum to exceed $\ 10^9$.
I was able to do part $a$, so I proved it for $n$. However, I actually don't know part $b$. How to solve an inequality with a factorial? Thank you for helping in advance.
Note that factorial function is an increasing function.
Hence solving $(n+1)!-1>10^9$ can be done for example by using bisection search.