Infinite series convergence and sum

6

1. Find as a single algebraic fraction an expression for \(\sum _ { r = 1 } ^ { n } \frac { 1 } { ( 2 r - 1 ) ( 2 r + 1 ) }\).
2. Determine the value of \(\sum _ { r = 1 } ^ { \infty } \frac { 1 } { ( 2 r - 1 ) ( 2 r + 1 ) }\).

7

1. Determine an expression for \(\sum _ { r = 1 } ^ { n } \frac { 1 } { r ( r + 1 ) ( r + 2 ) }\) giving your answer in the form \(\frac { 1 } { 4 } - \frac { 1 } { 2 } \mathrm { f } ( n )\).
2. Find the value of \(\sum _ { r = 1 } ^ { \infty } \frac { 1 } { r ( r + 1 ) ( r + 2 ) }\).

3

1. Find \(\sum _ { r = 1 } ^ { n } \left( \frac { 1 } { r } - \frac { 1 } { r + 2 } \right)\).
2. What does the sum in part (i) tend to as \(n \rightarrow \infty\) ? Justify your answer.

4 For each of the following improper integrals, find the value of the integral or explain briefly why it does not have a value:

1. \(\quad \int _ { 2 } ^ { \infty } 8 x ^ { - 3 } \mathrm {~d} x\);
   (3 marks)
2. \(\quad \int _ { 2 } ^ { \infty } \left( 8 x ^ { - 3 } + 1 \right) \mathrm { d } x\);
3. \(\quad \int _ { 2 } ^ { \infty } 8 x ^ { - 3 } ( x + 1 ) \mathrm { d } x\).

5

1. Explain why \(\int _ { 0 } ^ { \frac { 1 } { 16 } } x ^ { - \frac { 1 } { 2 } } \mathrm {~d} x\) is an improper integral.
2. For each of the following improper integrals, find the value of the integral or explain briefly why it does not have a value:
   1. \(\int _ { 0 } ^ { \frac { 1 } { 16 } } x ^ { - \frac { 1 } { 2 } } \mathrm {~d} x\);
   2. \(\int _ { 0 } ^ { \frac { 1 } { 16 } } x ^ { - \frac { 5 } { 4 } } \mathrm {~d} x\).

9

1. Show that, for all positive integers \(r\), $$\frac { r + 1 } { r + 2 } - \frac { r } { r + 1 } = \frac { 1 } { ( r + 1 ) ( r + 2 ) }$$ 9
2. Hence, using the method of differences, show that $$\sum _ { r = 1 } ^ { n } \frac { 1 } { ( r + 1 ) ( r + 2 ) } = \frac { n } { a n + b }$$ where \(a\) and \(b\) are integers to be determined.
   9
3. Hence find the exact value of $$\sum _ { r = 1001 } ^ { 2000 } \frac { 1 } { ( r + 1 ) ( r + 2 ) }$$ \(\_\_\_\_\) The curve \(C\) has equation $$y = \frac { 2 x - 10 } { 3 x - 5 }$$ Figure 1 shows the curve \(C\) with its asymptotes. \begin{figure}[h]
   \captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{47b12ae4-ca3f-472c-9d15-2ef17a2a4d87-12_979_1079_641_468}
   \end{figure}

4

1. Determine an expression for \(\sum _ { r = 1 } ^ { n } \frac { 1 } { r ( r + 1 ) ( r + 2 ) }\) giving your answer in the form \(\frac { 1 } { 4 } - \frac { 1 } { 2 } \mathrm { f } ( n )\).
2. Find the value of \(\sum _ { r = 1 } ^ { \infty } \frac { 1 } { r ( r + 1 ) ( r + 2 ) }\).

2 In this question you must show detailed reasoning.

1. Use partial fractions to show that \(\sum _ { r = 5 } ^ { n } \frac { 3 } { r ^ { 2 } + r - 2 } = \frac { 37 } { 60 } - \frac { 1 } { n } - \frac { 1 } { n + 1 } - \frac { 1 } { n + 2 }\).
2. Write down the value of \(\lim _ { n \rightarrow \infty } \left( \sum _ { r = 5 } ^ { n } \frac { 3 } { r ^ { 2 } + r - 2 } \right)\).

1. Show that \(\tanh(\ln n) = \frac{n^2 - 1}{n^2 + 1}\). [2]

It is given that, for non-negative integers \(n\), \(I_n = \int_0^{\ln 2} \tanh^n u du\).

1. Show that \(I_n - I_{n-2} = -\frac{1}{n-1}\left(\frac{3}{5}\right)^{n-1}\), for \(n \geq 2\). [3]
2. Find the value of \(I_3\), giving your answer in the form \(a + \ln b\), where \(a\) and \(b\) are constants. [4]
3. Use the method of differences on the result of part (ii) to find the sum of the infinite series $$\frac{1}{2}\left(\frac{3}{5}\right)^2 + \frac{1}{4}\left(\frac{3}{5}\right)^4 + \frac{1}{6}\left(\frac{3}{5}\right)^6 + \ldots.$$ [2]

Let \(J_n = \int_1^{\mathrm{e}} (\ln x)^n \, \mathrm{d}x\), where \(n\) is a positive integer. By considering \(\frac{\mathrm{d}}{\mathrm{d}x}(x(\ln x)^n)\), or otherwise, show that $$J_n = \mathrm{e} - nJ_{n-1}.$$ [4] Let \(J_n = \frac{J_n}{n!}\). Show that $$\frac{1}{2!} - \frac{1}{3!} + \frac{1}{4!} - \ldots + \frac{1}{10!} = \frac{1}{\mathrm{e}}(1 + J_{10}).$$ [6] It can be shown that $$\sum_{r=2}^{n} \frac{(-1)^r}{r!} = \frac{1}{\mathrm{e}}(1 + (-1)^n J_n)$$ for all positive integers \(n\). Deduce the sum to infinity of the series $$\frac{1}{2!} - \frac{1}{3!} + \frac{1}{4!} - \ldots,$$ justifying your conclusion carefully. [3]