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

Write it as a^n\left(1+\frac bax\right)^n and expand.

Range of validity for (a+bx)^n?

\left|\frac bax\right|<1, i.e. |x|<\left|\frac ab\right|.

Expansion of \frac{1}{(2+x)^2} to x^2?

\frac14-\frac x4+\frac{3x^2}{16}, valid for |x|<2.

Expansion of \sqrt[3]{8+3x} to x^2?

2+\frac x4-\frac{x^2}{32}, valid for |x|<\frac83.

How do you expand a rational function with two factors?

Write it in partial fractions, expand each, then add.

What is the validity range of a sum of expansions?

The overlap: values valid for every term, usually the smaller range.

Why must x lie inside the range of validity when approximating?

Outside it the series does not converge to the value of the function.

Binomial series for any rational powerEdexcel International A Level Maths: Flashcards

Card 1 of 120 of 12 known

Question

State the binomial series for $(1+x)^n$ with $n$ rational.

Tap or press Space to reveal

Tap card or press Space to flip

See all 12 cards

State the binomial series for $(1+x)^n$ with $n$ rational.: $1+nx+\frac{n(n-1)}{2!}x^2+\frac{n(n-1)(n-2)}{3!}x^3+\ldots$
When is $(1+x)^n$ valid for non-integer or negative $n$ ?: For $|x|<1$ .
How do you expand $(a+bx)^n$ ?: Write it as $a^n\left(1+\frac bax\right)^n$ and expand.
Range of validity for $(a+bx)^n$ ?: $\left|\frac bax\right|<1$ , i.e. $|x|<\left|\frac ab\right|$ .
Expand $(1+x)^{-1}$ .: $1-x+x^2-x^3+\ldots$ for $|x|<1$ .
Expand $(1+x)^{\frac12}$ to $x^2$ .: $1+\frac12x-\frac18x^2$
Expand $(1+4x)^{-\frac12}$ to $x^3$ .: $1-2x+6x^2-20x^3$
Expansion of $\frac{1}{(2+x)^2}$ to $x^2$ ?: $\frac14-\frac x4+\frac{3x^2}{16}$ , valid for $|x|<2$ .
Expansion of $\sqrt[3]{8+3x}$ to $x^2$ ?: $2+\frac x4-\frac{x^2}{32}$ , valid for $|x|<\frac83$ .
How do you expand a rational function with two factors?: Write it in partial fractions, expand each, then add.
What is the validity range of a sum of expansions?: The overlap: values valid for every term, usually the smaller range.
Why must x lie inside the range of validity when approximating?: Outside it the series does not converge to the value of the function.

Exam questions on Binomial series for any rational power

The function $f(x)=(1+4x)^{-\frac12}$ is expanded as a series in ascending powers of $x$ .
Find the term in $x^3$ in the expansion of $f(x)$ .2 marks
The function $g(x)=\frac{1}{(2+x)^2}$ is expanded as a series in ascending powers of $x$ .
Use the expansion of $g(x)$ up to and including the term in $x^2$ , with $x=0.1$ , to estimate $\frac{1}{2.1^2}$ to 4 decimal places.2 marks
The function $h(x)=\sqrt[3]{8+3x}$ is expanded as a series in ascending powers of $x$ .
Find the expansion of $h(x)$ in ascending powers of $x$ up to and including the term in $x^2$ , giving each coefficient as a fraction in its simplest form.3 marks

See the full worksheet

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).