sequences series geometric series inequality irrationality proof proof by contradiction factorial exponential function integer part function comparison test

Problem

No solution available for this problem.

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*}
\begin{questionparts}
\item Evaluate $b_n$.
\item Show that $0<a_n<1/n$.
\item Deduce that $a_n=n!{\rm e}-[n!{\rm e}]$ (where $[x]$ is
the integer part of $x$).
\item Hence show that $\mathrm{e}$ is irrational.
\end{questionparts}