4．A sequence of positive integers \(a _ { 1 } , a _ { 2 } , a _ { 3 } , \ldots\) has \(r\) th term given by $$a _ { r } = 2 ^ { r } - 1$$ （a）Write down the first 6 terms of this sequence．
（b）Verify that \(a _ { r + 1 } = 2 a _ { r } + 1\)
（c）Find \(\sum _ { r = 1 } ^ { n } a _ { r }\)
（d）Show that \(\frac { 1 } { a _ { r + 1 } } < \frac { 1 } { 2 } \times \frac { 1 } { a _ { r } }\)
（e）Hence show that \(1 + \frac { 1 } { 3 } + \frac { 1 } { 7 } + \frac { 1 } { 15 } + \frac { 1 } { 31 } + \ldots < 1 + \frac { 1 } { 3 } + \left( \frac { 1 } { 7 } + \frac { \frac { 1 } { 2 } } { 7 } + \frac { \frac { 1 } { 4 } } { 7 } + \ldots \right)\)
（f）Show that \(\frac { 31 } { 21 } < \sum _ { r = 1 } ^ { \infty } \frac { 1 } { a _ { r } } < \frac { 34 } { 21 }\)