6 You are given that \(q \in \mathbb { Z }\) with \(q \geqslant 1\) and that
\(\mathrm { S } = \frac { 1 } { ( \mathrm { q } + 1 ) } + \frac { 1 } { ( \mathrm { q } + 1 ) ( \mathrm { q } + 2 ) } + \frac { 1 } { ( \mathrm { q } + 1 ) ( \mathrm { q } + 2 ) ( \mathrm { q } + 3 ) } + \ldots\).

1. By considering a suitable geometric series show that \(\mathrm { S } < \frac { 1 } { \mathrm { q } }\).
2. Deduce that \(S \notin \mathbb { Z }\). You are also given that \(\mathrm { e } = \sum _ { r = 0 } ^ { \infty } \frac { 1 } { r ! }\).
3. Assume that \(\mathrm { e } = \frac { \mathrm { p } } { \mathrm { q } }\), where \(p\) and \(q\) are positive integers. By writing the infinite series for e in a form using \(q\) and \(S\) and using the result from part (b), prove by contradiction that e is irrational.