Partial fractions with verification

A question is this type if and only if it asks to express a function in partial fractions and then verify/show that a given series expansion is correct when higher powers are neglected.

3 questions · Standard +0.3

1.02y Partial fractions: decompose rational functions
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CAIE P3 2003 June Q6
9 marks Standard +0.3
6 Let \(\mathrm { f } ( x ) = \frac { 9 x ^ { 2 } + 4 } { ( 2 x + 1 ) ( x - 2 ) ^ { 2 } }\).
  1. Express \(\mathrm { f } ( x )\) in partial fractions.
  2. Show that, when \(x\) is sufficiently small for \(x ^ { 3 }\) and higher powers to be neglected, $$f ( x ) = 1 - x + 5 x ^ { 2 }$$
CAIE P3 2004 June Q9
9 marks Standard +0.3
9 Let \(\mathrm { f } ( x ) = \frac { x ^ { 2 } + 7 x - 6 } { ( x - 1 ) ( x - 2 ) ( x + 1 ) }\).
  1. Express \(\mathrm { f } ( x )\) in partial fractions.
  2. Show that, when \(x\) is sufficiently small for \(x ^ { 4 }\) and higher powers to be neglected, $$f ( x ) = - 3 + 2 x - \frac { 3 } { 2 } x ^ { 2 } + \frac { 11 } { 4 } x ^ { 3 } .$$
CAIE P3 2002 November Q6
9 marks Standard +0.3
6 Let \(f ( x ) = \frac { 6 + 7 x } { ( 2 - x ) \left( 1 + x ^ { 2 } \right) }\).
  1. Express \(\mathrm { f } ( x )\) in partial fractions.
  2. Show that, when \(x\) is sufficiently small for \(x ^ { 4 }\) and higher powers to be neglected, $$f ( x ) = 3 + 5 x - \frac { 1 } { 2 } x ^ { 2 } - \frac { 15 } { 4 } x ^ { 3 }$$