When does the binomial expansion stop?

Only when n is a positive integer; otherwise it is an infinite series.

How do you expand (a+bx)^n when a\ne1?

Take out a^n: a^n\left(1+\frac bax\right)^n, then expand.

Range of validity of the expansion of (a+bx)^n?

\left|\frac{bx}{a}\right|<1, i.e. |x|<\left|\frac ab\right|

What is (9-2x)^{-\frac12} in the form a^n(1+\ldots)^n?

\frac13\left(1-\frac{2x}{9}\right)^{-\frac12}

How do you estimate a number such as \sqrt{3.96} using the expansion?

Choose x so that (4+x)^{\frac12} gives it (x=-0.04), check it is in the range, then substitute.

Why check the range of validity before substituting a value?

Outside the range the series does not converge to the function, so the estimate is meaningless.

How do you expand a fraction like \frac{5-x}{(1+x)(1-2x)}?

Split into partial fractions, expand each part, then add the series.

Range of validity for a sum of two expansions?

The values where both are valid, i.e. the more restrictive range.

Binomial expansion for rational powersEdexcel A-Level Maths: Flashcards

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Binomial expansion of $(1+x)^n$ for rational $n$?

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Binomial expansion of $(1+x)^n$ for rational $n$ ?: $1+nx+\frac{n(n-1)}{2!}x^2+\frac{n(n-1)(n-2)}{3!}x^3+\ldots$ , valid for $|x|<1$ .
When does the binomial expansion stop?: Only when $n$ is a positive integer; otherwise it is an infinite series.
How do you expand $(a+bx)^n$ when $a\ne1$ ?: Take out $a^n$ : $a^n\left(1+\frac bax\right)^n$ , then expand.
Range of validity of the expansion of $(a+bx)^n$ ?: $\left|\frac{bx}{a}\right|<1$ , i.e. $|x|<\left|\frac ab\right|$
Expand $(1+x)^{-1}$ to three terms.: $1-x+x^2$
Expand $(1+x)^{\frac12}$ to three terms.: $1+\frac12x-\frac18x^2$
Expand $(1-x)^{-2}$ to three terms.: $1+2x+3x^2$
What is $(9-2x)^{-\frac12}$ in the form $a^n(1+\ldots)^n$ ?: $\frac13\left(1-\frac{2x}{9}\right)^{-\frac12}$
How do you estimate a number such as $\sqrt{3.96}$ using the expansion?: Choose $x$ so that $(4+x)^{\frac12}$ gives it ( $x=-0.04$ ), check it is in the range, then substitute.
Why check the range of validity before substituting a value?: Outside the range the series does not converge to the function, so the estimate is meaningless.
How do you expand a fraction like $\frac{5-x}{(1+x)(1-2x)}$ ?: Split into partial fractions, expand each part, then add the series.
Range of validity for a sum of two expansions?: The values where both are valid, i.e. the more restrictive range.

Exam questions on Binomial expansion for rational powers

$f(x)=(1+4x)^{-\frac12}$
Find the coefficient of $x^2$ in the expansion of $f(x)$ .2 marks
$g(x)=\sqrt{4+x}$
Find the first three terms of the expansion of $g(x)$ in ascending powers of $x$ .2 marks
$h(x)=\frac{1}{\sqrt{9-2x}}$
Find the first three terms of the binomial expansion of $h(x)$ in ascending powers of $x$ .3 marks

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Written by the Exaim team, led by Shaun Daswani (Head of Upper Secondary, Improve ME Institute; MSc Financial Mathematics, Imperial College London; BSc, UCL) and Jason Daswani (operational lead, Improve ME Institute; LSE).