Series expansion of rational function - Generalised Binomial Theorem

Edexcel C4 2007 January Q1

5 marks Moderate -0.3

1. $$f ( x ) = ( 2 - 5 x ) ^ { - 2 } , \quad | x | < \frac { 2 } { 5 }$$ Find the binomial expansion of $\mathrm { f } ( x )$, in ascending powers of $x$, as far as the term in $x ^ { 3 }$, giving each coefficient as a simplified fraction.
(5)

Edexcel C4 2013 January Q1

5 marks Moderate -0.8

Given

$$f ( x ) = ( 2 + 3 x ) ^ { - 3 } , \quad | x | < \frac { 2 } { 3 }$$ find the binomial expansion of $\mathrm { f } ( x )$, in ascending powers of $x$, up to and including the term in $x ^ { 3 }$. Give each coefficient as a simplified fraction.

Edexcel C4 2007 June Q1

5 marks Moderate -0.3

1. $$f ( x ) = ( 3 + 2 x ) ^ { - 3 } , \quad | x | < \frac { 3 } { 2 }$$ Find the binomial expansion of $\mathrm { f } ( x )$, in ascending powers of $x$, as far as the term in $x ^ { 3 }$.
Give each coefficient as a simplified fraction.
(5)

Edexcel C4 2016 June Q1

6 marks Moderate -0.3

Use the binomial series to find the expansion of

$$\frac { 1 } { ( 2 + 5 x ) ^ { 3 } } , \quad | x | < \frac { 2 } { 5 }$$ in ascending powers of $x$, up to and including the term in $x ^ { 3 }$.
Give each coefficient as a fraction in its simplest form.
(6)

Edexcel P4 2018 Specimen Q1

6 marks Moderate -0.3

Use the binomial series to find the expansion of

$$\frac { 1 } { ( 2 + 5 x ) ^ { 3 } } \quad | \boldsymbol { x } | < \frac { 2 } { 5 }$$ in ascending powers of $x$, up to and including the term in $x ^ { 3 }$ Give each coefficient as a fraction in its simplest form.

VIIIV SIHI NI JIIIM ION OC

VIIV SIHI NI JAHAM ION OO

VJ4V SIHIL NI JIIIM IONOO

OCR MEI C4 Q1

5 marks Moderate -0.5

1 Find the coefficient of the term in $x ^ { 3 }$ in the expansion of $\frac { 1 } { ( 2 + 3 x ) ^ { 2 } }$.

OCR MEI C4 2011 January Q3

7 marks Moderate -0.3

3 Find the first three terms in the binomial expansion of $\frac { 1 } { ( 3 - 2 x ) ^ { 3 } }$ in ascending powers of $x$. State the set of values of $x$ for which the expansion is valid.

Edexcel AEA 2004 June Q2

10 marks Challenging +1.3

For the binomial expansion of $\frac{1}{(1-x)^2}$, $|x| < 1$, in ascending powers of $x$,
1. find the first four terms,
2. write down the coefficient of $x^n$. [2]
Hence, show that, for $|x| < 1$, $\sum_{n=1}^{\infty} nx^n = \frac{x}{(1-x)^2}$. [2]
Prove that, for $|x| < 1$, $\sum_{n=1}^{\infty} (an+1)x^n = \frac{(a+1)x-x^2}{(1-x)^2}$, where $a$ is a constant. [4]
Hence evaluate $\sum_{n=1}^{\infty} \frac{5n+1}{2^{3n}}$. [2]