Factorial or product method of differences

8

1. Show that \(( r + 2 ) ! - ( r + 1 ) ! = ( r + 1 ) ^ { 2 } \times r !\).
2. Hence find an expression, in terms of \(n\), for $$2 ^ { 2 } \times 1 ! + 3 ^ { 2 } \times 2 ! + 4 ^ { 2 } \times 3 ! + \ldots + ( n + 1 ) ^ { 2 } \times n ! .$$
3. State, giving a brief reason, whether the series $$2 ^ { 2 } \times 1 ! + 3 ^ { 2 } \times 2 ! + 4 ^ { 2 } \times 3 ! + \ldots$$ converges.

3

1. Show that \(\frac { 1 } { r ! } - \frac { 1 } { ( r + 1 ) ! } = \frac { r } { ( r + 1 ) ! }\).
2. Hence find an expression, in terms of \(n\), for $$\frac { 1 } { 2 ! } + \frac { 2 } { 3 ! } + \frac { 3 } { 4 ! } + \ldots + \frac { n } { ( n + 1 ) ! }$$

1 Let \(\mathrm { f } ( r ) = r ! ( r - 1 )\). Simplify \(\mathrm { f } ( r + 1 ) - \mathrm { f } ( r )\) and hence find \(\sum _ { r = n + 1 } ^ { 2 n } r ! \left( r ^ { 2 } + 1 \right)\).

3 Verify that if $$v _ { n } = n ( n + 1 ) ( n + 2 ) \ldots ( n + m )$$ then $$v _ { n + 1 } - v _ { n } = ( m + 1 ) ( n + 1 ) ( n + 2 ) \ldots ( n + m ) .$$ Given now that $$u _ { n } = ( n + 1 ) ( n + 2 ) \ldots ( n + m ) ,$$ find \(\sum _ { n = 1 } ^ { N } u _ { n }\) in terms of \(m\) and \(N\).

1

1. Express \(\frac { 1 } { ( r + 2 ) r ! }\) in the form \(\frac { A } { ( r + 1 ) ! } + \frac { B } { ( r + 2 ) ! }\), where \(A\) and \(B\) are integers.
   [0pt] [3 marks]
2. Hence find \(\sum _ { r = 1 } ^ { n } \frac { 1 } { ( r + 2 ) r ! }\).
   [0pt] [2 marks]

11

1. Show that, for all positive integers \(r\), $$\frac { 1 } { ( r - 1 ) ! } - \frac { 1 } { r ! } = \frac { r - 1 } { r ! }$$ ⟶
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2. Hence, using the method of differences, show that $$\sum _ { r = 1 } ^ { n } \frac { r - 1 } { r ! } = a + \frac { b } { n ! }$$ where \(a\) and \(b\) are integers to be determined.