For each positive integer \(n\),
let
\begin{align*}
a_n&=\frac1{n+1}+\frac1{(n+1)(n+2)}+\frac1{(n+1)(n+2)(n+3)}+\cdots;\\
b_n&=\frac1{n+1}+\frac1{(n+1)^2}+\frac1{(n+1)^3}+\cdots.
\end{align*}
Evaluate \(b_n\).
Show that \(0
Deduce that \(a_n=n!{\rm e}-[n!{\rm e}]\) (where \([x]\) is
the integer part of \(x\)).