Method of differences with exponential/logarithmic terms

4 Let \(\mathrm { u } _ { \mathrm { r } } = \mathrm { e } ^ { \mathrm { rx } } \left( \mathrm { e } ^ { 2 \mathrm { x } } - 2 \mathrm { e } ^ { \mathrm { x } } + 1 \right)\).

1. Using the method of differences, or otherwise, find \(\sum _ { \mathrm { r } = 1 } ^ { \mathrm { n } } \mathrm { u } _ { \mathrm { r } }\) in terms of \(n\) and \(x\).
2. Deduce the set of non-zero values of \(x\) for which the infinite series $$u _ { 1 } + u _ { 2 } + u _ { 3 } + \ldots$$ is convergent and give the sum to infinity when this exists.
3. Using a standard result from the list of formulae (MF19), find \(\sum _ { \mathrm { r } = 1 } ^ { \mathrm { n } } \ln \mathrm { u } _ { \mathrm { r } }\) in terms of \(n\) and \(x\).

3 Let \(S _ { n } = \sum _ { r = 1 } ^ { n } \ln \frac { r ( r + 2 ) } { ( r + 1 ) ^ { 2 } }\).

1. Using the method of differences, or otherwise, show that \(S _ { n } = \ln \frac { n + 2 } { 2 ( n + 1 ) }\).
   Let \(S = \sum _ { r = 1 } ^ { \infty } \ln \frac { r ( r + 2 ) } { ( r + 1 ) ^ { 2 } }\).
2. Find the least value of \(n\) such that \(\mathrm { S } _ { \mathrm { n } } - \mathrm { S } < 0.01\).

2 Given that $$u _ { n } = \ln \left( \frac { 1 + x ^ { n + 1 } } { 1 + x ^ { n } } \right)$$ where \(x > - 1\), find \(\sum _ { n = 1 } ^ { N } u _ { n }\) in terms of \(N\) and \(x\). Find the sum to infinity of the series $$u _ { 1 } + u _ { 2 } + u _ { 3 } + \ldots$$ when

1. \(- 1 < x < 1\),
2. \(x = 1\).

2

1. Verify that $$\frac { n ( \mathrm { e } - 1 ) + \mathrm { e } } { n ( n + 1 ) \mathrm { e } ^ { n + 1 } } = \frac { 1 } { n \mathrm { e } ^ { n } } - \frac { 1 } { ( n + 1 ) \mathrm { e } ^ { n + 1 } }$$ Let \(S _ { N } = \sum _ { n = 1 } ^ { N } \frac { n ( \mathrm { e } - 1 ) + \mathrm { e } } { n ( n + 1 ) \mathrm { e } ^ { n + 1 } }\).
2. Express \(S _ { N }\) in terms of \(N\) and e.
   Let \(S = \lim _ { N \rightarrow \infty } S _ { N }\).
3. Find the least value of \(N\) such that \(( N + 1 ) \left( S - S _ { N } \right) < 10 ^ { - 3 }\).

1 Given that $$u _ { n } = \mathrm { e } ^ { n x } - \mathrm { e } ^ { ( n + 1 ) x }$$ find \(\sum _ { n = 1 } ^ { N } \| _ { n }\) in terms of \(N\) and \(x\). Hence determine the set of values of \(x\) for which the infinite series $$u _ { 1 } + u _ { 2 } + u _ { 3 } + \ldots$$ is convergent and give the sum to infinity for cases where this exists.

1. (i) Find the value of

$$\sum _ { r = 4 } ^ { \infty } 20 \times \left( \frac { 1 } { 2 } \right) ^ { r }$$ (3)
(ii) Show that $$\sum _ { n = 1 } ^ { 48 } \log _ { 5 } \left( \frac { n + 2 } { n + 1 } \right) = 2$$

3

1. Show that $$\frac { 2 ^ { r + 1 } } { r + 2 } - \frac { 2 ^ { r } } { r + 1 } = \frac { r 2 ^ { r } } { ( r + 1 ) ( r + 2 ) }$$
2. Hence find $$\sum _ { r = 1 } ^ { 30 } \frac { r 2 ^ { r } } { ( r + 1 ) ( r + 2 ) }$$ giving your answer in the form \(2 ^ { n } - 1\), where \(n\) is an integer.

1. (a) Use the method of differences to prove that for \(n > 2\)

$$\sum _ { r = 2 } ^ { n } \ln \left( \frac { r + 1 } { r - 1 } \right) \equiv \ln \left( \frac { n ( n + 1 ) } { 2 } \right)$$ (4)
(b) Hence find the exact value of $$\sum _ { r = 51 } ^ { 100 } \ln \left( \frac { r + 1 } { r - 1 } \right) ^ { 35 }$$ Give your answer in the form \(a \ln \left( \frac { b } { c } \right)\) where \(a , b\) and \(c\) are integers to
be determined.

9

1. Show that, for \(r > 0\), $$\ln ( r + 2 ) - \ln r = \ln \left( 1 + \frac { 2 } { r } \right)$$ 9
2. Hence, using the method of differences, show that $$\sum _ { r = 1 } ^ { n } \ln \left( 1 + \frac { 2 } { r } \right) = \ln \left( \frac { 1 } { 2 } ( n + a ) ( n + b ) \right)$$ where \(a\) and \(b\) are integers to be found.

5 The function f is defined by $$f ( r ) = 2 ^ { r } ( r - 2 ) \quad ( r \in \mathbb { Z } )$$ 5

1. Show that $$\mathrm { f } ( r + 1 ) - \mathrm { f } ( r ) = r 2 ^ { r }$$ 5
2. Use the method of differences to show that $$\sum _ { r = 1 } ^ { n } r 2 ^ { r } = 2 ^ { n + 1 } ( n - 1 ) + 2$$