Infinite series convergence and sum

A question is this type if and only if it asks to deduce the value of Ī£(r=1 to āˆž) by taking the limit as nā†’āˆž of a finite sum, or to determine convergence conditions.

37 questions · Standard +0.6

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AQA FP2 2013 January Q3
7 marks Standard +0.3
3
  1. Show that \(\frac { 1 } { 5 r - 2 } - \frac { 1 } { 5 r + 3 } = \frac { A } { ( 5 r - 2 ) ( 5 r + 3 ) }\), stating the value of the constant \(A\).
    (2 marks)
  2. Hence use the method of differences to show that $$\sum _ { r = 1 } ^ { n } \frac { 1 } { ( 5 r - 2 ) ( 5 r + 3 ) } = \frac { n } { 3 ( 5 n + 3 ) }$$
  3. Find the value of $$\sum _ { r = 1 } ^ { \infty } \frac { 1 } { ( 5 r - 2 ) ( 5 r + 3 ) }$$ (1 mark)
OCR MEI Further Pure Core 2023 June Q3
6 marks Standard +0.8
3
  1. Using partial fractions and the method of differences, show that $$\frac { 1 } { 1 \times 3 } + \frac { 1 } { 2 \times 4 } + \frac { 1 } { 3 \times 5 } + \ldots + \frac { 1 } { \mathrm { n } ( \mathrm { n } + 2 ) } = \frac { 3 } { 4 } - \frac { \mathrm { an } + \mathrm { b } } { 2 ( \mathrm { n } + 1 ) ( \mathrm { n } + 2 ) }$$ where \(a\) and \(b\) are integers to be determined.
  2. Deduce the sum to infinity of the series. $$\frac { 1 } { 1 \times 3 } + \frac { 1 } { 2 \times 4 } + \frac { 1 } { 3 \times 5 } + \ldots$$
CAIE FP1 2019 November Q5
9 marks Challenging +1.8
  1. Use standard results from the List of Formulae (MF10) to show that $$S _ { N } = \frac { 1 } { 3 } N \left( 25 N ^ { 2 } + 90 N + 83 \right)$$
  2. Use the method of differences to express \(T _ { N }\) in terms of \(N\).
  3. Find \(\lim _ { N \rightarrow \infty } \left( N ^ { - 3 } S _ { N } T _ { N } \right)\).
OCR FP1 2016 June Q8
10 marks Standard +0.3
  1. Show that \(\frac { 1 } { 2 r + 1 } - \frac { 1 } { 2 r + 3 } \equiv \frac { 2 } { ( 2 r + 1 ) ( 2 r + 3 ) }\).
  2. Hence find \(\sum _ { r = 1 } ^ { n } \frac { 1 } { ( 2 r + 1 ) ( 2 r + 3 ) }\), giving your answer as a single fraction.
  3. Find \(\sum _ { r = n } ^ { \infty } \frac { 1 } { ( 2 r + 1 ) ( 2 r + 3 ) }\), giving your answer as a single fraction.
OCR Further Pure Core 1 2018 September Q6
5 marks Standard +0.8
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 \(\sum _ { r = 1 } ^ { \infty } \frac { 1 } { ( 2 r - 1 ) ( 2 r + 1 ) }\).
OCR Further Pure Core 1 2018 December Q7
7 marks Challenging +1.2
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 \(\sum _ { r = 1 } ^ { \infty } \frac { 1 } { r ( r + 1 ) ( r + 2 ) }\).
OCR Further Pure Core 2 2017 Specimen Q3
4 marks Standard +0.3
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.
AQA FP1 2005 January Q4
7 marks Standard +0.3
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. \(\quad \int _ { 2 } ^ { \infty } 8 x ^ { - 3 } ( x + 1 ) \mathrm { d } x\).
AQA FP1 2010 January Q5
7 marks Standard +0.3
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. \(\int _ { 0 } ^ { \frac { 1 } { 16 } } x ^ { - \frac { 5 } { 4 } } \mathrm {~d} x\).
AQA Further AS Paper 1 2024 June Q9
7 marks Standard +0.3
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}
OCR Further Pure Core 1 2021 June Q4
7 marks Challenging +1.2
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 \(\sum _ { r = 1 } ^ { \infty } \frac { 1 } { r ( r + 1 ) ( r + 2 ) }\).
OCR Further Pure Core 2 2021 June Q2
6 marks Standard +0.3
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 \(\lim _ { n \rightarrow \infty } \left( \sum _ { r = 5 } ^ { n } \frac { 3 } { r ^ { 2 } + r - 2 } \right)\).