4.06b Method of differences: telescoping series

2 By considering the identity $$\cos [ ( 2 n - 1 ) \alpha ] - \cos [ ( 2 n + 1 ) \alpha ] \equiv 2 \sin \alpha \sin 2 n \alpha$$ show that if \(\alpha\) is not an integer multiple of \(\pi\) then $$\sum _ { n = 1 } ^ { N } \sin ( 2 n \alpha ) = \frac { 1 } { 2 } \cot \alpha - \frac { 1 } { 2 } \operatorname { cosec } \alpha \cos [ ( 2 N + 1 ) \alpha ]$$ Deduce that the infinite series $$\sum _ { n = 1 } ^ { \infty } \sin \left( \frac { 2 } { 3 } n \pi \right)$$ does not converge.

1 Express \(\frac { 1 } { ( 2 r + 1 ) ( 2 r + 3 ) }\) in partial fractions and hence use the method of differences to find $$\sum _ { r = 1 } ^ { n } \frac { 1 } { ( 2 r + 1 ) ( 2 r + 3 ) }$$ Deduce the value of $$\sum _ { r = 1 } ^ { \infty } \frac { 1 } { ( 2 r + 1 ) ( 2 r + 3 ) }$$

3 Given that \(\mathrm { f } ( r ) = \frac { 1 } { ( r + 1 ) ( r + 2 ) }\), show that $$\mathrm { f } ( r - 1 ) - \mathrm { f } ( r ) = \frac { 2 } { r ( r + 1 ) ( r + 2 ) }$$ Hence find \(\sum _ { r = 1 } ^ { n } \frac { 1 } { r ( r + 1 ) ( r + 2 ) }\). Deduce the value of \(\sum _ { r = 1 } ^ { \infty } \frac { 1 } { r ( r + 1 ) ( r + 2 ) }\).

1 Find the sum of the first \(n\) terms of the series $$\frac { 1 } { 1 \times 3 } + \frac { 1 } { 2 \times 4 } + \frac { 1 } { 3 \times 5 } + \ldots$$ and deduce the sum to infinity.

5 Use the method of differences to show that \(\sum _ { r = 1 } ^ { N } \frac { 1 } { ( 2 r + 1 ) ( 2 r + 3 ) } = \frac { 1 } { 6 } - \frac { 1 } { 2 ( 2 N + 3 ) }\). Deduce that \(\sum _ { r = N + 1 } ^ { 2 N } \frac { 1 } { ( 2 r + 1 ) ( 2 r + 3 ) } < \frac { 1 } { 8 N }\).

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

2 Expand and simplify \(( r + 1 ) ^ { 4 } - r ^ { 4 }\). Use the method of differences together with the standard results for \(\sum _ { r = 1 } ^ { n } r\) and \(\sum _ { r = 1 } ^ { n } r ^ { 2 }\) to show that $$\sum _ { r = 1 } ^ { n } r ^ { 3 } = \frac { 1 } { 4 } n ^ { 2 } ( n + 1 ) ^ { 2 }$$

2 Express \(\frac { 4 } { r ( r + 1 ) ( r + 2 ) }\) in partial fractions and hence find \(\sum _ { r = 1 } ^ { n } \frac { 4 } { r ( r + 1 ) ( r + 2 ) }\). Deduce the value of \(\sum _ { r = 1 } ^ { \infty } \frac { 4 } { r ( r + 1 ) ( r + 2 ) }\).

1 It is given that \(\sum _ { r = 1 } ^ { n } u _ { r } = n ^ { 2 } ( 2 n + 3 )\), where \(n\) is a positive integer.

1. Find \(\sum _ { r = n + 1 } ^ { 2 n } u _ { r }\).
2. Find \(u _ { r }\).

2

1. Verify that \(\frac { 2 r + 1 } { r ( r + 1 ) ( r + 2 ) } = \frac { 1 } { 2 } \left\{ \frac { ( 2 r + 1 ) ( 2 r + 3 ) } { ( r + 1 ) ( r + 2 ) } - \frac { ( 2 r - 1 ) ( 2 r + 1 ) } { r ( r + 1 ) } \right\}\).
2. Hence show that \(\sum _ { r = 1 } ^ { n } \frac { 2 r + 1 } { r ( r + 1 ) ( r + 2 ) } = \frac { 1 } { 2 } \left\{ \frac { ( 2 n + 1 ) ( 2 n + 3 ) } { ( n + 1 ) ( n + 2 ) } - \frac { 3 } { 2 } \right\}\).
3. Deduce the value of \(\sum _ { r = 1 } ^ { \infty } \frac { 2 r + 1 } { r ( r + 1 ) ( r + 2 ) }\).

5 Let \(S _ { n } = \sum _ { r = 1 } ^ { n } ( - 1 ) ^ { r - 1 } r ^ { 2 }\).

1. Use the standard result for \(\sum _ { r = 1 } ^ { n } r ^ { 2 }\) given in the List of Formulae (MF10) to show that $$S _ { 2 n } = - n ( 2 n + 1 )$$
2. State the value of \(\lim _ { n \rightarrow \infty } \frac { S _ { 2 n } } { n ^ { 2 } }\) and find \(\lim _ { n \rightarrow \infty } \frac { S _ { 2 n + 1 } } { n ^ { 2 } }\).

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 \(( N + 1 ) \left( S - S _ { N } \right) < 10 ^ { - 3 }\).

9 For the sequence \(u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots\), it is given that \(u _ { 1 } = 8\) and $$u _ { r + 1 } = \frac { 5 u _ { r } - 3 } { 4 }$$ for all \(r\).

1. Prove by mathematical induction that $$u _ { n } = 4 \left( \frac { 5 } { 4 } \right) ^ { n } + 3$$ for all positive integers \(n\).
2. Deduce the set of values of \(x\) for which the infinite series $$\left( u _ { 1 } - 3 \right) x + \left( u _ { 2 } - 3 \right) x ^ { 2 } + \ldots + \left( u _ { r } - 3 \right) x ^ { r } + \ldots$$ is convergent.
3. Use the result given in part (i) to find surds \(a\) and \(b\) such that $$\sum _ { n = 1 } ^ { N } \ln \left( u _ { n } - 3 \right) = N ^ { 2 } \ln a + N \ln b .$$

2 Let \(u _ { n } = \frac { 4 \sin \left( n - \frac { 1 } { 2 } \right) \sin \frac { 1 } { 2 } } { \cos ( 2 n - 1 ) + \cos 1 }\).

1. Using the formulae for \(\cos P \pm \cos Q\) given in the List of Formulae MF10, show that $$u _ { n } = \frac { 1 } { \cos n } - \frac { 1 } { \cos ( n - 1 ) }$$
2. Use the method of differences to find \(\sum _ { n = 1 } ^ { N } u _ { n }\).
3. Explain why the infinite series \(u _ { 1 } + u _ { 2 } + u _ { 3 } + \ldots\) does not converge.

4

1. Use the method of differences to show that \(\sum _ { r = 1 } ^ { N } \frac { 1 } { ( 3 r + 1 ) ( 3 r - 2 ) } = \frac { 1 } { 3 } - \frac { 1 } { 3 ( 3 N + 1 ) }\).
2. Find the limit, as \(N \rightarrow \infty\), of \(\sum _ { r = N + 1 } ^ { N ^ { 2 } } \frac { N } { ( 3 r + 1 ) ( 3 r - 2 ) }\).

The positive variables \(y\) and \(t\) are related by $$y = a ^ { t }$$ where \(a\) is a positive constant.

1. (a) By differentiating \(\ln y\) with respect to \(t\), show that \(\frac { \mathrm { d } y } { \mathrm {~d} t } = a ^ { t } \ln a\).
   (b) Write down \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} t ^ { 2 } }\).
2. Determine the set of values of \(a\) for which the infinite series $$y + \frac { \mathrm { d } y } { \mathrm {~d} t } + \frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} t ^ { 2 } } + \frac { \mathrm { d } ^ { 3 } y } { \mathrm {~d} t ^ { 3 } } + \ldots$$ is convergent.
   A curve has parametric equations $$x = t ^ { a } , \quad y = a ^ { t }$$
3. Find \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\) in terms of \(a\) and \(t\), and show that, when \(t = 2\), $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } = 2 ^ { 1 - 2 a } ( 1 - a + 2 \ln a ) \ln a$$ If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

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.

5 Let $$S _ { N } = \sum _ { n = 1 } ^ { N } ( - 1 ) ^ { n - 1 } n ^ { 3 }$$ Find \(S _ { 2 N }\) in terms of \(N\), simplifying your answer as far as possible. Hence write down an expression for \(S _ { 2 N + 1 }\) and find the limit, as \(N \rightarrow \infty\), of \(\frac { S _ { 2 N + 1 } } { N ^ { 3 } }\).

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 \(N\).

9 Use induction to prove that $$\sum _ { n = 1 } ^ { N } \frac { 4 n + 1 } { n ( n + 1 ) ( 2 n - 1 ) ( 2 n + 1 ) } = 1 - \frac { 1 } { ( N + 1 ) ( 2 N + 1 ) }$$ Show that $$\sum _ { n = N + 1 } ^ { 2 N } \frac { 4 n + 1 } { n ( n + 1 ) ( 2 n - 1 ) ( 2 n + 1 ) } < \frac { 3 } { 8 N ^ { 2 } }$$

2 Use the method of differences to find \(S _ { N }\), where $$S _ { N } = \sum _ { n = 1 } ^ { N } \frac { 1 } { n ( n + 2 ) }$$ Deduce the value of \(\lim _ { N \rightarrow \infty } S _ { N }\).

1 Verify that \(\frac { 1 } { n ^ { 2 } } - \frac { 1 } { ( n + 1 ) ^ { 2 } } = \frac { 2 n + 1 } { n ^ { 2 } ( n + 1 ) ^ { 2 } }\). Let \(S _ { N } = \sum _ { r = 1 } ^ { N } \frac { 2 r + 1 } { r ^ { 2 } ( r + 1 ) ^ { 2 } }\). Express \(S _ { N }\) in terms of \(N\). Let \(S = \lim _ { N \rightarrow \infty } S _ { N }\). Find the least value of \(N\) such that \(S - S _ { N } < 10 ^ { - 16 }\).

4 Let \(\mathrm { f } ( r ) = r ( r + 1 ) ( r + 2 )\). Show that $$\mathrm { f } ( r ) - \mathrm { f } ( r - 1 ) = 3 r ( r + 1 )$$ Hence show that \(\sum _ { r = 1 } ^ { n } r ( r + 1 ) = \frac { 1 } { 3 } n ( n + 1 ) ( n + 2 )\). Using the standard result for \(\sum _ { r = 1 } ^ { n } r\), deduce that \(\sum _ { r = 1 } ^ { n } r ^ { 2 } = \frac { 1 } { 6 } n ( n + 1 ) ( 2 n + 1 )\). Find the sum of the series $$1 ^ { 2 } + 2 \times 2 ^ { 2 } + 3 ^ { 2 } + 2 \times 4 ^ { 2 } + 5 ^ { 2 } + 2 \times 6 ^ { 2 } + \ldots + 2 ( n - 1 ) ^ { 2 } + n ^ { 2 }$$ where \(n\) is odd.

The roots of the equation \(x ^ { 4 } - 3 x ^ { 2 } + 5 x - 2 = 0\) are \(\alpha , \beta , \gamma , \delta\), and \(\alpha ^ { n } + \beta ^ { n } + \gamma ^ { n } + \delta ^ { n }\) is denoted by \(S _ { n }\). Show that $$S _ { n + 4 } - 3 S _ { n + 2 } + 5 S _ { n + 1 } - 2 S _ { n } = 0$$ Find the values of

1. \(S _ { 2 }\) and \(S _ { 4 }\),
2. \(S _ { 3 }\) and \(S _ { 5 }\). Hence find the value of $$\alpha ^ { 2 } \left( \beta ^ { 3 } + \gamma ^ { 3 } + \delta ^ { 3 } \right) + \beta ^ { 2 } \left( \gamma ^ { 3 } + \delta ^ { 3 } + \alpha ^ { 3 } \right) + \gamma ^ { 2 } \left( \delta ^ { 3 } + \alpha ^ { 3 } + \beta ^ { 3 } \right) + \delta ^ { 2 } \left( \alpha ^ { 3 } + \beta ^ { 3 } + \gamma ^ { 3 } \right) .$$

1 Express \(\frac { 1 } { r ( r + 1 ) ( r - 1 ) }\) in partial fractions. Find $$\sum _ { r = 2 } ^ { n } \frac { 1 } { r ( r + 1 ) ( r - 1 ) }$$ State the value of $$\sum _ { r = 2 } ^ { \infty } \frac { 1 } { r ( r + 1 ) ( r - 1 ) }$$