Binomial Expansion III: Fractional & Negative
IB Mathematics AA · HL · Shadow Worksheet
Practice
Question 1
Find the first four terms of the following expansions, stating the range of validity:
(a)\((1+x)^{-2}\)
(b)\((1-3x)^{1/2}\)
(c)\((1+2x)^{-1/3}\)
(a)
Answer
\(1-2x+3x^2-4x^3+\ldots,\quad|x|<1\)
(b)
Use \((1+u)^{1/2}\approx1+\tfrac{1}{2}u-\tfrac{1}{8}u^2+\tfrac{1}{16}u^3\) with \(u=-3x\)
Answer
\(1-\tfrac{3}{2}x-\tfrac{9}{8}x^2-\tfrac{27}{16}x^3+\ldots,\quad|x|<\tfrac{1}{3}\)
(c)
Use \(n=-\tfrac{1}{3}\), \(u=2x\)
\(1+(-\tfrac{1}{3})(2x)+\dfrac{(-\tfrac{1}{3})(-\tfrac{4}{3})}{2}(2x)^2+\dfrac{(-\tfrac{1}{3})(-\tfrac{4}{3})(-\tfrac{7}{3})}{6}(2x)^3+\ldots\)
Answer
\(1-\tfrac{2}{3}x+\tfrac{8}{9}x^2-\tfrac{112}{81}x^3+\ldots,\quad|x|<\tfrac{1}{2}\)
Question 2
Expand \(\dfrac{1}{(1+x)^3}\) as a series in ascending powers of \(x\) up to and including \(x^3\). State the range of validity.
Write as
\((1+x)^{-3}=1-3x+6x^2-10x^3+\ldots\)
Answer
\(1-3x+6x^2-10x^3+\ldots,\quad|x|<1\)
Question 3
Use the expansion of \((1+x)^{1/2}\) to show that \(\sqrt{1.08}\approx1.0392\) using two terms.
Substitute \(x=0.08\)
\((1+0.08)^{1/2}\approx1+\tfrac{1}{2}(0.08)=1.04\); using three terms: \(1+0.04-\tfrac{1}{8}(0.08)^2=1.0392\)
Answer
Shown: \(\sqrt{1.08}\approx1.0392\)
Question 4
Expand \(\sqrt{\dfrac{1-x}{1+x}}\) up to and including the term in \(x^2\).
Write as
\((1-x)^{1/2}(1+x)^{-1/2}\approx\!\left(1-\tfrac{x}{2}-\tfrac{x^2}{8}\right)\!\!\left(1-\tfrac{x}{2}+\tfrac{3x^2}{8}\right)\)
Expand to \(x^2\)
Constant: 1; \(x\): \(-\tfrac{1}{2}-\tfrac{1}{2}=-1\); \(x^2\): \(-\tfrac{1}{8}+\tfrac{3}{8}+\tfrac{1}{4}=\tfrac{1}{2}\)
Answer
\(1-x+\tfrac{x^2}{2}+\ldots,\quad|x|<1\)
Question 5
Find the range of values of \(x\) for which the expansion of \(\dfrac{1}{\sqrt{4-3x}}\) is valid, and find the first three terms.
Rewrite
\(\dfrac{1}{2}\!\left(1-\tfrac{3x}{4}\right)^{-1/2}\); valid when \(\left|\tfrac{3x}{4}\right|<1\Rightarrow|x|<\tfrac{4}{3}\)
Expand
\(\tfrac{1}{2}\!\left(1+\tfrac{3x}{8}+\tfrac{27x^2}{128}+\ldots\right)\)
Answer
\(\tfrac{1}{2}+\tfrac{3x}{16}+\tfrac{27x^2}{256}+\ldots,\quad|x|<\tfrac{4}{3}\)
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