Repeated linear factor with series expansion

Denominator includes a repeated linear factor (squared term); requires expanding terms like (1±ax)^(-2) using binomial series in addition to (1±ax)^(-1).

4 questions

Edexcel C4 Q7

7. $$f ( x ) = \frac { 25 } { ( 3 + 2 x ) ^ { 2 } ( 1 - x ) } , \quad | x | < 1$$

Express \(\mathrm { f } ( x )\) as a sum of partial fractions.
Hence find \(\int f ( x ) d x\).
Find the series expansion of \(\mathrm { f } ( x )\) in ascending powers of \(x\) up to and including the term in \(x ^ { 2 }\). Give each coefficient as a simplified fraction. END

Edexcel C4 Q5

5. $$f ( x ) = \frac { 5 - 8 x } { ( 1 + 2 x ) ( 1 - x ) ^ { 2 } }$$

Express \(\mathrm { f } ( x )\) in partial fractions.
Find the series expansion of \(\mathrm { f } ( x )\) in ascending powers of \(x\) up to and including the term in \(x ^ { 3 }\), simplifying each coefficient.
State the set of values of \(x\) for which your expansion is valid.
5. continued

SPS SPS FM Pure 2024 January Q5

5. Let $$f ( x ) = \frac { 27 x ^ { 2 } + 32 x + 16 } { ( 3 x + 2 ) ^ { 2 } ( 1 - x ) }$$

Express \(f ( x )\) in terms of partial fractions
Hence, or otherwise, find the series expansion of \(\mathrm { f } ( x )\), in ascending powers of \(x\), up to and including the term in \(x ^ { 2 }\). Simplify each term.
State, with a reason, whether your series expansion in part (b) is valid for \(x = \frac { 1 } { 2 }\).
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OCR Pure 1 2018 March Q9

Express \(\frac { 5 + 4 x - 3 x ^ { 2 } } { ( 1 - 2 x ) ( 2 + x ) ^ { 2 } }\) in three partial fractions.
Hence find the first three terms in the expansion of \(\frac { 5 + 4 x - 3 x ^ { 2 } } { ( 1 - 2 x ) ( 2 + x ) ^ { 2 } }\) in ascending powers of \(x\).
State the set of values for which the expansion in part (ii) is valid.