Alternating series summation

A question is this type if and only if it involves summing a series with alternating signs like Σ(-1)^r f(r) or separating odd and even terms.

7 questions · Standard +0.5

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CAIE FP1 2011 June Q1
5 marks Standard +0.3
1 Find \(2 ^ { 2 } + 4 ^ { 2 } + \ldots + ( 2 n ) ^ { 2 }\). Hence find \(1 ^ { 2 } - 2 ^ { 2 } + 3 ^ { 2 } - 4 ^ { 2 } + \ldots - ( 2 n ) ^ { 2 }\), simplifying your answer.
CAIE FP1 2018 June Q5
8 marks Standard +0.8
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 \(\lim _ { n \rightarrow \infty } \frac { S _ { 2 n + 1 } } { n ^ { 2 } }\).
CAIE FP1 2004 November Q5
7 marks Challenging +1.2
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 \(\frac { S _ { 2 N + 1 } } { N ^ { 3 } }\).
WJEC Further Unit 1 2023 June Q10
8 marks Challenging +1.2
10. Gareth is investigating a series involving cube numbers. His series is $$1 ^ { 3 } - 2 ^ { 3 } + 3 ^ { 3 } - 4 ^ { 3 } + 5 ^ { 3 } - 6 ^ { 3 } + 7 ^ { 3 } - \ldots$$ Gareth continues his series and ends with an odd number.
Find and simplify an expression for the sum of Gareth's series in terms of \(k\), where \(k\) is the number of odd numbers in his series.
Edexcel CP1 2023 June Q7
12 marks Standard +0.8
  1. In this question you must show all stages of your working. Solutions relying on calculator technology are not acceptable.
    1. Explain why, for \(n \in \mathbb { N }\)
    $$\sum _ { r = 1 } ^ { 2 n } ( - 1 ) ^ { r } \mathrm { f } ( r ) = \sum _ { r = 1 } ^ { n } ( \mathrm { f } ( 2 r ) - \mathrm { f } ( 2 r - 1 ) )$$ for any function \(\mathrm { f } ( r )\).
  2. Use the standard summation formulae to show that, for \(n \in \mathbb { N }\) $$\sum _ { r = 1 } ^ { 2 n } r \left( ( - 1 ) ^ { r } + 2 r \right) ^ { 2 } = n ( 2 n + 1 ) \left( 8 n ^ { 2 } + 4 n + 5 \right)$$
  3. Hence evaluate $$\sum _ { r = 14 } ^ { 50 } r \left( ( - 1 ) ^ { r } + 2 r \right) ^ { 2 }$$
Pre-U Pre-U 9794/2 2013 June Q8
4 marks Moderate -0.8
8 Evaluate the following, giving your answers in exact form.
  1. \(\sum _ { n = 1 } ^ { 30 } \frac { 1 } { n } - \sum _ { n = 2 } ^ { 29 } \frac { 1 } { n }\).
  2. \(\sum _ { n = 1 } ^ { 100 } n \times ( - 1 ) ^ { n }\).
Pre-U Pre-U 9795 Specimen Q4
6 marks Standard +0.3
Write down the sum $$\sum_{n=1}^{2N} n^3$$ in terms of \(N\), and hence find $$1^3 - 2^3 + 3^3 - 4^3 + \ldots - (2N)^3$$ in terms of \(N\), simplifying your answer. [6]