It is conjectured that
$$\frac { 1 } { 2 ! } + \frac { 2 } { 3 ! } + \frac { 3 } { 4 ! } + \ldots + \frac { n - 1 } { n ! } = a - \frac { b } { n ! } ,$$
where \(a\) and \(b\) are constants, and \(n\) is an integer such that \(n \geq 2\).
By considering particular cases, show that if the conjecture is correct then \(a = b = 1\).
Use induction to prove that
$$\frac { 1 } { 2 ! } + \frac { 2 } { 3 ! } + \frac { 3 } { 4 ! } + \ldots + \frac { n - 1 } { n ! } = 1 - \frac { 1 } { n ! } \text { for } n \geq 2 .$$