1.04d Binomial expansion validity: convergence conditions

10

1. Show that \(\frac { x } { ( 1 - x ) ^ { 3 } } \approx x + 3 x ^ { 2 } + 6 x ^ { 3 }\) for small values of \(x\).
2. Use this result, together with a suitable value of \(x\), to obtain a decimal estimate of the value of \(\frac { 100 } { 729 }\).
3. Show that \(\frac { x } { ( 1 - x ) ^ { 3 } } = - \frac { 1 } { x ^ { 2 } } \left( 1 - \frac { 1 } { x } \right) ^ { - 3 }\). Hence find the first three terms of the binomial expansion of \(\frac { x } { ( 1 - x ) ^ { 3 } }\) in powers of \(\frac { 1 } { x }\).
4. Comment on the suitability of substituting the same value of \(x\) as used in part (ii) in the expansion in part (iii) to estimate the value of \(\frac { 100 } { 729 }\).

3

1. Find the first three terms in the expansion of \(( 1 - 2 x ) ^ { - \frac { 1 } { 2 } }\) in ascending powers of \(x\), where \(| x | < \frac { 1 } { 2 }\).
2. Hence find the coefficient of \(x ^ { 2 }\) in the expansion of \(\frac { x + 3 } { \sqrt { 1 - 2 x } }\).

4

1. Find the first three terms in the binomial expansion of \(( 8 - 9 x ) ^ { \frac { 2 } { 3 } }\) in ascending powers of \(x\).
2. State the set of values of \(x\) for which this expansion is valid.

7 Given that the binomial expansion of \(( 1 + k x ) ^ { n }\) is \(1 - 6 x + 30 x ^ { 2 } + \ldots\), find the values of \(n\) and \(k\). State the set of values of \(x\) for which this expansion is valid.

2 Show that \(( 1 + 2 x ) ^ { \frac { 1 } { 3 } } = 1 + \frac { 2 } { 3 } x - \frac { 4 } { 9 } x ^ { 2 } + \ldots\), and find the next term in the expansion.
State the set of values of \(x\) for which the expansion is valid.

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

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.

2 Find the first four terms of the binomial expansion of \(\sqrt [ 3 ] { 1 - 2 x }\). State the set of values of \(x\) for which the expansion is valid.

3

1. Find the first three terms of the binomial expansion of \(\frac { 1 } { \sqrt [ 3 ] { 1 - 2 x } }\). State the set of values of \(x\) for which
   the expansion is valid. the expansion is valid.
2. Hence find \(a\) and \(b\) such that \(\frac { 1 - 3 x } { \sqrt [ 3 ] { 1 - 2 x } } = 1 + a x + b x ^ { 2 } + \ldots\).

2 Find the first 4 terms in the binomial expansion of \(\sqrt { 4 + 2 x }\). State the range of values of \(x\) for which the expansion is valid.

8

1. Find the first three terms in the expansion of \(( 4 - x ) ^ { - \frac { 1 } { 2 } }\) in ascending powers of \(x\).
2. The expansion of \(\frac { a + b x } { \sqrt { 4 - x } }\) is \(16 - x \ldots\). Find the values of the constants \(a\) and \(b\).

6

1. Find the first three terms in the expansion of \(( 8 - 3 x ) ^ { \frac { 1 } { 3 } }\) in ascending powers of \(x\).
2. State the range of values of \(x\) for which the expansion in part (a) is valid.
3. Find the coefficient of \(x ^ { 2 }\) in the expansion of \(\frac { ( 8 - 3 x ) ^ { \frac { 1 } { 3 } } } { ( 1 + 2 x ) ^ { 2 } }\).

8

1. Find the first three terms in the expansion of \(( 4 + 3 x ) ^ { \frac { 3 } { 2 } }\) in ascending powers of \(x\).
2. State the range of values of \(x\) for which the expansion in part (a) is valid.
3. In the expansion of \(( 4 + 3 x ) ^ { \frac { 3 } { 2 } } ( 1 + a x ) ^ { 2 }\) the coefficient of \(x ^ { 2 }\) is \(\frac { 107 } { 16 }\). Determine the possible values of the constant \(a\).

2

1. Find the coefficient of \(x ^ { 5 }\) in the expansion of \(( 3 - 2 x ) ^ { 8 }\).
2. 1. Expand \(( 1 + 3 x ) ^ { 0.5 }\) as far as the term in \(x ^ { 3 }\).
   2. State the range of values of \(x\) for which your expansion is valid. A student suggests the following check to determine whether the expansion obtained in part (b)(i) may be correct.
      "Use the expansion to find an estimate for \(\sqrt { 103 }\), correct to five decimal places, and compare this with the value of \(\sqrt { 103 }\) given by your calculator."
   3. Showing your working, carry out this check on your expansion from part (b)(i).

1. The binomial expansion of

$$( 1 + a x ) ^ { 12 }$$ up to and including the term in \(x ^ { 2 }\) is $$1 - \frac { 15 } { 2 } x + k x ^ { 2 }$$ where \(a\) and \(k\) are constants.

1. Show that \(a = - \frac { 5 } { 8 }\)
2. Hence find the value of \(k\) Using the expansion and making your method clear,
3. find an estimate for the value of \(\left( \frac { 17 } { 16 } \right) ^ { 12 }\), giving your answer to 4 decimal places.

1. a. Find the first four terms, in ascending powers of \(x\), of the binomial expansion of

$$\left( \frac { 1 } { 9 } - 2 x \right) ^ { \frac { 1 } { 2 } }$$ giving each coefficient in its simplest form.
b. Explain how you could use \(x = \frac { 1 } { 36 }\) in the expansion to find an approximation for \(\sqrt { 2 }\). There is no need to carry out the calculation.

7. a. Use the binomial theorem to expand $$( 8 - 3 x ) ^ { \frac { 2 } { 3 } }$$ in ascending powers of \(x\), up to and including the term \(x ^ { 3 }\), as a fully simplifying each term. Edward, a student decides to use the expansion with \(x = \frac { 1 } { 3 }\) to find an approximation for \(( 7 ) ^ { \frac { 2 } { 3 } }\). Using the answer to part (a) and without doing any calculations, b. explain clearly whether Edward's approximation will be an overestimate, or, an underestimate.

1. (a) Use binomial expansions to show that \(\sqrt { \frac { 1 + 4 x } { 1 - x } } \approx 1 + \frac { 5 } { 2 } x - \frac { 5 } { 8 } x ^ { 2 }\)

A student substitutes \(x = \frac { 1 } { 2 }\) into both sides of the approximation shown in part (a) in an attempt to find an approximation to \(\sqrt { 6 }\) (b) Give a reason why the student should not use \(x = \frac { 1 } { 2 }\) (c) Substitute \(x = \frac { 1 } { 11 }\) into $$\sqrt { \frac { 1 + 4 x } { 1 - x } } = 1 + \frac { 5 } { 2 } x - \frac { 5 } { 8 } x ^ { 2 }$$ to obtain an approximation to \(\sqrt { 6 }\). Give your answer as a fraction in its simplest form.

1. (a) Find the first three terms, in ascending powers of \(x\), of the binomial expansion of

$$\frac { 1 } { \sqrt { 4 - x } }$$ giving each coefficient in its simplest form. The expansion can be used to find an approximation to \(\sqrt { 2 }\) Possible values of \(x\) that could be substituted into this expansion are:

• \(x = - 14\) because \(\frac { 1 } { \sqrt { 4 - x } } = \frac { 1 } { \sqrt { 18 } } = \frac { \sqrt { 2 } } { 6 }\)
• \(x = 2\) because \(\frac { 1 } { \sqrt { 4 - x } } = \frac { 1 } { \sqrt { 2 } } = \frac { \sqrt { 2 } } { 2 }\)
• \(x = - \frac { 1 } { 2 }\) because \(\frac { 1 } { \sqrt { 4 - x } } = \frac { 1 } { \sqrt { \frac { 9 } { 2 } } } = \frac { \sqrt { 2 } } { 3 }\) (b) Without evaluating your expansion,
  1. state, giving a reason, which of the three values of \(x\) should not be used
  2. state, giving a reason, which of the three values of \(x\) would lead to the most accurate approximation to \(\sqrt { 2 }\)

1. (a) Find, in ascending powers of \(x\), the first four terms of the binomial expansion of

$$( 1 - 9 x ) ^ { \frac { 1 } { 2 } }$$ giving each term in simplest form.
(b) Give a reason why \(x = - \frac { 2 } { 9 }\) should not be used in the expansion to find an approximation to \(\sqrt { 3 }\)

1. (a) Show that the binomial expansion of

$$( 4 + 5 x ) ^ { \frac { 1 } { 2 } }$$ in ascending powers of \(x\), up to and including the term in \(x ^ { 2 }\) is $$2 + \frac { 5 } { 4 } x + k x ^ { 2 }$$ giving the value of the constant \(k\) as a simplified fraction.
(b) (i) Use the expansion from part (a), with \(x = \frac { 1 } { 10 }\), to find an approximate value for \(\sqrt { 2 }\) Give your answer in the form \(\frac { p } { q }\) where \(p\) and \(q\) are integers.
(ii) Explain why substituting \(x = \frac { 1 } { 10 }\) into this binomial expansion leads to a valid approximation.

1. (a) Find the first four terms, in ascending powers of \(x\), of the binomial expansion of

$$\sqrt { 4 - 9 x }$$ writing each term in simplest form. A student uses this expansion with \(x = \frac { 1 } { 9 }\) to find an approximation for \(\sqrt { 3 }\) Using the answer to part (a) and without doing any calculations,
(b) state whether this approximation will be an overestimate or an underestimate of \(\sqrt { 3 }\) giving a brief reason for your answer.

1. (a) Use the binomial expansion, in ascending powers of \(x\), to show that

$$\sqrt { ( 4 - x ) } = 2 - \frac { 1 } { 4 } x + k x ^ { 2 } + \ldots$$ where \(k\) is a rational constant to be found. A student attempts to substitute \(x = 1\) into both sides of this equation to find an approximate value for \(\sqrt { 3 }\).
(b) State, giving a reason, if the expansion is valid for this value of \(x\).

13 The function \(\mathrm { f } ( x )\) is defined by \(\mathrm { f } ( x ) = \sqrt [ 3 ] { 27 - 8 x ^ { 3 } }\). Jenny uses her scientific calculator to create a table of values for \(\mathrm { f } ( x )\) and \(\mathrm { f } ^ { \prime } ( x )\).

1. Use calculus to find an expression for \(\mathrm { f } ^ { \prime } ( x )\) and hence explain why the calculator gives an error for \(\mathrm { f } ^ { \prime } ( 1.5 )\).
2. Find the first three terms of the binomial expansion of \(\mathrm { f } ( x )\).
3. Jenny integrates the first three terms of the binomial expansion of \(\mathrm { f } ( x )\) to estimate the value of \(\int _ { 0 } ^ { 1 } \sqrt [ 3 ] { 27 - 8 x ^ { 3 } } \mathrm {~d} x\). Explain why Jenny's method is valid in this case. (You do not need to evaluate Jenny's approximation.)
4. Use the trapezium rule with 4 strips to obtain an estimate for \(\int _ { 0 } ^ { 1 } \sqrt [ 3 ] { 27 - 8 x ^ { 3 } } \mathrm {~d} x\). The calculator gives 2.92117438 for \(\int _ { 0 } ^ { 1 } \sqrt [ 3 ] { 27 - 8 x ^ { 3 } } \mathrm {~d} x\). The graph of \(y = \mathrm { f } ( x )\) is shown in Fig. 13. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{904025c9-6d68-4344-bd41-8c0fccfcf92f-08_490_906_1505_568} \captionsetup{labelformat=empty} \caption{Fig. 13}
   \end{figure}
5. Explain why the trapezium rule gives an underestimate.

3 The function \(\mathrm { f } ( x )\) is given by \(\mathrm { f } ( x ) = ( 1 - a x ) ^ { - 3 }\), where \(a\) is a non-zero constant. In the binomial expansion of \(\mathrm { f } ( x )\), the coefficients of \(x\) and \(x ^ { 2 }\) are equal.

1. Find the value of \(a\).
2. Using this value for \(a\),
   1. state the set of values of \(x\) for which the binomial expansion is valid,
   2. write down the quadratic function which approximates \(\mathrm { f } ( x )\) when \(x\) is small.