Partial fractions for summation

A question is this type if and only if it uses partial fractions to evaluate or simplify a summation series, typically involving telescoping sums.

2 questions · Standard +0.8

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OCR Further Pure Core 2 2020 November Q3
6 marks Standard +0.8
3 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 \(\lim _ { n \rightarrow \infty } \left( \sum _ { r = 5 } ^ { n } \frac { 3 } { r ^ { 2 } + r - 2 } \right)\).
CAIE P3 2024 November Q11
14 marks Standard +0.8
Let \(f(x) = \frac{2e^{2x}}{e^{2x} - 3e^x + 2}\).
  1. Find \(f'(x)\) and hence find the exact coordinates of the stationary point of the curve with equation \(y = f(x)\). [5]
  2. Use the substitution \(u = e^x\) and partial fractions to find the exact value of \(\int_{\ln 5} f(x) dx\). Give your answer in the form \(\ln a\), where \(a\) is a rational number in its simplest form. [9]