1.04d Binomial expansion validity: convergence conditions

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1. Find the first three terms in the expansion of \(( 1 + p x ) ^ { \frac { 1 } { 3 } }\) in ascending powers of \(x\).
2. The expansion of \(( 1 + q x ) ( 1 + p x ) ^ { \frac { 1 } { 3 } }\) is \(1 + x - \frac { 2 } { 9 } x ^ { 2 } + \ldots\). Find the possible values of the constants \(p\) and \(q\).

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1. Find the first three terms, in ascending powers of \(x\), of the binomial expansion of \(\frac { 1 } { \sqrt { 4 + x } }\) 6
2. Hence, find the first three terms of the binomial expansion of \(\frac { 1 } { \sqrt { 4 - x ^ { 3 } } }\) 6 (d) (i) 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 (d) (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.

1 The first three terms, in ascending powers of \(x\), of the binomial expansion of \(( 9 + 2 x ) ^ { \frac { 1 } { 2 } }\) are given by $$( 9 + 2 x ) ^ { \frac { 1 } { 2 } } \approx a + \frac { x } { 3 } - \frac { x ^ { 2 } } { 54 }$$ where \(a\) is a constant. 1

1. State the range of values of \(x\) for which this expansion is valid.
   Circle your answer. \(| x | < \frac { 2 } { 9 }\) \(| x | < \frac { 2 } { 3 }\) \(| x | < 1\) \(| x | < \frac { 9 } { 2 }\) 1
2. Find the value of \(a\).
   Circle your answer.
   [0pt] [1 mark]
   1239

9

1. Find the first three terms, in ascending powers of \(x\), of the binomial expansion of $$( 1 + x ) ^ { - \frac { 1 } { 2 } }$$ 9
2. A student substitutes \(x = 2\) into the expansion of \(( 1 + x ) ^ { - \frac { 1 } { 2 } }\) to find an approximation for \(\frac { 1 } { \sqrt { 3 } }\) Explain the mistake in the student's approach.
   [0pt] [1 mark] 9
3. By substituting \(x = - \frac { 1 } { 4 }\) in your expansion for \(( 1 + x ) ^ { - \frac { 1 } { 2 } }\) find an approximation for \(\frac { 1 } { \sqrt { 3 } }\) Give your answer to three significant figures.

1. (a) Find the first 4 terms of the binomial expansion, in ascending powers of \(x\), of

$$( 8 - 3 x ) ^ { - \frac { 1 } { 3 } } \quad | x | < \frac { 8 } { 3 }$$ giving each coefficient as a simplified fraction.
(b) Use the answer from part (a) with \(x = \frac { 2 } { 3 }\) to find a rational approximation to \(\sqrt [ 3 ] { 6 }\)

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1. Show that the first three terms in the expansion of \(( 1 - 2 x ) ^ { \frac { 1 } { 2 } }\) are \(1 - x - \frac { 1 } { 2 } x ^ { 2 }\) and find the next term.
2. State the range of values of \(x\) for which this expansion is valid.
3. Hence show that the first four terms in the expansion of \(( 2 + x ) ( 1 - 2 x ) ^ { \frac { 1 } { 2 } }\) are \(2 - x + a x ^ { 2 } + b x ^ { 3 }\) and state the values of \(a\) and \(b\).

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1. Expand \(( 1 + x ) ^ { - 1 }\) up to and including the term in \(x ^ { 2 }\).
2. (a) Expand \(\sqrt { 2 + 3 x ^ { 2 } }\) up to and including the term in \(x ^ { 4 }\).
   (b) For what range of values of \(x\) is this expansion valid?
3. Find the first three terms of the expansion of \(\frac { \sqrt { 2 + 3 x ^ { 2 } } } { 1 + x }\) in ascending powers of \(x\) and hence show that \(\int _ { 0 } ^ { 0.1 } \frac { \sqrt { 2 + 3 x ^ { 2 } } } { 1 + x } \mathrm {~d} x \approx 0.135\).

Expand \(\frac { 4 - x } { \sqrt { 1 + 2 x } }\) in ascending powers of \(x\) up to and including the term in \(x ^ { 3 }\). State the range of values of \(x\) for which the expansion is valid.

Find the first three terms in the binomial expansion of \(\frac { 2 - x } { \sqrt { 1 + 3 x } }\) in ascending powers of \(x\). State the range of values of \(x\) for which the expansion is valid. By writing \(x = \frac { 1 } { 22 }\) in your expansion, find an approximate value for \(\sqrt { 22 }\) in the form \(\frac { a } { b }\), where \(a , b\) are integers whose values are to be found.

Given that the binomial expansion of \((1 + kx)^{-4}\), \(|kx| < 1\), is $$1 - 6x + Ax^2 + \ldots$$

1. [(a)] find the value of the constant \(k\), \hfill [2]
2. [(b)] find the value of the constant \(A\), giving your answer in its simplest form. \hfill [3]

$$f(x) = (8 - 3x)^{\frac{4}{3}} \quad 0 < x < \frac{8}{3}$$

1. Show that the binomial expansion of \(f(x)\) in ascending powers of \(x\) up to and including the term in \(x^3\) is $$A - 8x + \frac{x^2}{2} + Bx^3 + ...$$ where \(A\) and \(B\) are constants to be found. [4]
2. Use proof by contradiction to prove that the curve with equation $$y = 8 + 8x - \frac{15}{2}x^2$$ does not intersect the curve with equation $$y = A - 8x + \frac{x^2}{2} + Bx^3 \quad 0 < x < \frac{8}{3}$$ where \(A\) and \(B\) are the constants found in part (a). (Solutions relying on calculator technology are not acceptable.) [4]

$$g(x) = \frac{1}{\sqrt{4-x^2}}$$

1. Find, in ascending powers of \(x\), the first four non-zero terms of the binomial expansion of \(g(x)\). Give each coefficient in simplest form. [5]
2. State the range of values of \(x\) for which this expansion is valid. [1]
3. Use the expansion from part (a) to find a fully simplified rational approximation for \(\sqrt{3}\) Show your working and make your method clear. [2]

Given the binomial expansion \((1 + qx)^p = 1 - x + 2x^2 + \ldots\), find the values of \(p\) and \(q\). Hence state the set of values of \(x\) for which the expansion is valid. [6]

Find the first three terms in the binomial expansion of \(\sqrt{1 + 3x}\) in ascending powers of \(x\). State the set of values of \(x\) for which the expansion is valid. [5]

Find the first four terms in the binomial expansion of \(\sqrt{1+2x}\). State the set of values of \(x\) for which the expansion is valid. [5]

Find the first three terms in the binomial expansion of \((4+x)^{\frac{1}{2}}\). State the set of values of \(x\) for which the expansion is valid. [5]

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

1. Express \(f(x)\) in partial fractions. [5]
2. Find the series expansion of \(f(x)\) in ascending powers of \(x\) up to and including the term in \(x^2\), simplifying each coefficient. [6]
3. State the set of values of \(x\) for which your expansion is valid. [1]

$$f(x) = 3 - \frac{x-1}{x-3} + \frac{x+11}{2x^2-5x-3}, \quad |x| < \frac{1}{2}.$$

1. Show that $$f(x) = \frac{4x-1}{2x+1}.$$ [4]
2. Find the series expansion of \(f(x)\) in ascending powers of \(x\) up to and including the term in \(x^3\), simplifying each coefficient. [5]