Substitute expression for variable - Generalised Binomial Theorem

4. $$f ( x ) = \frac { 27 } { ( 3 - 5 x ) ^ { 2 } } \quad | x | < \frac { 3 } { 5 }$$

Find the series expansion of $\mathrm { f } ( x )$, in ascending powers of $x$, up to and including the term in $x ^ { 3 }$. Give each coefficient in its simplest form.
(5) Use your answer to part (a) to find the series expansion in ascending powers of $x$, up to and including the term in $x ^ { 3 }$, of
$g ( x ) = \frac { 27 } { ( 3 + 5 x ) ^ { 2 } } \quad | x | < \frac { 3 } { 5 }$
$\mathrm { h } ( x ) = \frac { 27 } { ( 3 - x ) ^ { 2 } } \quad | x | < 3$

3. $$f ( x ) = \frac { 6 } { \sqrt { ( 9 - 4 x ) } } , \quad | x | < \frac { 9 } { 4 }$$

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 in its simplest form.
(6) Use your answer to part (a) to find the binomial expansion in ascending powers of $x$, up to and including the term in $x ^ { 3 }$, of
$\quad \mathrm { g } ( x ) = \frac { 6 } { \sqrt { } ( 9 + 4 x ) } , \quad | x | < \frac { 9 } { 4 }$
$\mathrm { h } ( x ) = \frac { 6 } { \sqrt { } ( 9 - 8 x ) } , \quad | x | < \frac { 9 } { 8 }$

5

Expand $( 1 - 3 x ) ^ { - \frac { 1 } { 3 } }$ in ascending powers of $x$, up to and including the term in $x ^ { 3 }$.
Hence find the coefficient of $x ^ { 3 }$ in the expansion of $\left( 1 - 3 \left( x + x ^ { 3 } \right) \right) ^ { - \frac { 1 } { 3 } }$.

3

Obtain the binomial expansion of $( 1 + x ) ^ { \frac { 1 } { 2 } }$ up to and including the term in $x ^ { 2 }$.
Hence obtain the binomial expansion of $\sqrt { 1 + \frac { 3 } { 2 } x }$ up to and including the term in $x ^ { 2 }$.
Hence show that $\sqrt { \frac { 2 + 3 x } { 8 } } \approx a + b x + c x ^ { 2 }$ for small values of $x$, where $a , b$ and $c$ are constants to be found.

3

1. Find the binomial expansion of $( 1 + x ) ^ { - \frac { 1 } { 3 } }$ up to and including the term in $x ^ { 2 }$.
  (2 marks)
2. Hence find the binomial expansion of $\left( 1 + \frac { 3 } { 4 } x \right) ^ { - \frac { 1 } { 3 } }$ up to and including the term in $x ^ { 2 }$.
Hence show that $\sqrt [ 3 ] { \frac { 256 } { 4 + 3 x } } \approx a + b x + c x ^ { 2 }$ for small values of $x$, stating the values of the constants $a , b$ and $c$.

4

Find the binomial expansion of $( 1 + x ) ^ { - \frac { 1 } { 2 } }$ up to the term in $x ^ { 2 }$.
Hence, or otherwise, obtain the binomial expansion of $\frac { 1 } { \sqrt { 1 + 2 x } }$ up to the term in $x ^ { 2 }$, in simplified form.
Use your answer to part (b) with $x = - 0.1$ to show that $\sqrt { 5 } \approx 2.23$.

2

Obtain the binomial expansion of $( 1 - x ) ^ { - 3 }$ up to and including the term in $x ^ { 2 }$.
Hence obtain the binomial expansion of $\left( 1 - \frac { 5 } { 2 } x \right) ^ { - 3 }$ up to and including the term in $x ^ { 2 }$.
Find the range of values of $x$ for which the binomial expansion of $\left( 1 - \frac { 5 } { 2 } x \right) ^ { - 3 }$ would be valid.
Given that $x$ is small, show that $\left( \frac { 4 } { 2 - 5 x } \right) ^ { 3 } \approx a + b x + c x ^ { 2 }$, where $a , b$ and $c$ are integers.

2 marks

6

Find the first three terms, in ascending powers of $x$, of the binomial expansion of $\frac { 1 } { \sqrt { 4 + x } }$
6
Hence, find the first three terms of the binomial expansion of $\frac { 1 } { \sqrt { 4 - x ^ { 3 } } }$
6
1. Edward, a student, decides to use this method to find a more accurate value for the integral by increasing the number of terms of the binomial expansion used. Explain clearly whether Edward's approximation will be an overestimate, an underestimate, or if it is impossible to tell.
  [0pt] [2 marks]
  6
(ii) Edward goes on to use the expansion from part (b) to find an approximation for $\int _ { - 2 } ^ { 0 } \frac { 1 } { \sqrt { 4 - x ^ { 3 } } } \mathrm {~d} x$ Explain why Edward's approximation is invalid.