Revise
Infinite Series Binomial Expansions
Infinite Series Binomial Expansions
For (a+bx)^{n}, we can still get an expansion if n is not a positive whole number. However, the expansion goes on forever. In this page you will find out how to calculate the expansion and how to use it.
Make sure you are happy with the following topics before continuing.
A Level Maths Predicted Papers 2025
The MME A level maths predicted papers are an excellent way to practise, using authentic exam style questions that are unique to our papers. Our examiners have studied A level maths past papers to develop predicted A level maths exam questions in an authentic exam format. The profit from every pack is reinvested into making free content on MME, which benefits millions of learners across the country.
View Product
A-A* A Level Maths Practice Papers
The MME A-A* A level maths practice papers are excellent for those top achieving students to practise for their exams, using authentic exam style questions that are unique to our practice papers. Our content experts have studied A level maths past papers and specifications to develop specific A-A* A level maths exam questions in an authentic exam style. The profit from every pack is reinvested into making free content on MME, which benefits millions of learners across the country.
View Product
The Infinite Binomial Expansion
Consider writing the binomial coefficients in a different way.
\begin{pmatrix}n\\0\end{pmatrix}=1
\begin{pmatrix}n\\1\end{pmatrix}=\dfrac{n}{1}
\begin{pmatrix}n\\2\end{pmatrix}=\dfrac{n(n-1)}{1\times2}
\begin{pmatrix}n\\3\end{pmatrix}=\dfrac{n(n-1)(n-2)}{1\times2\times3}
A clear pattern has emerged. Indeed:
\begin{pmatrix}n\\r\end{pmatrix}=\dfrac{n(n-1)(n-2)...(n-r+1)}{1\times2\times3\times...\times r}We can use this pattern instead of actual binomial coefficients to write an infinite expansion for (1+ax)^{n} when n is not a positive whole number.
(1+ax)^{n}=1+nax+\dfrac{n(n-1)}{1\times2}a^{2}x^{2}+...\dfrac{n(n-1)...(n-r+1)}{1\times2\times3\times...\times r}a^{r}x^{r}+...
Note: Factorisation
The formula above only works for expressions of the form (1+ax)^{n}, so how do we expand (ax+b)^{n}?
(ax+b)^{n}=b^{n}\left(1+\dfrac{a}{b}x\right)^{n}, and we can use our formula on \left(1+\dfrac{a}{b}x\right)^{n} then multiply by b^{n}.
Validity of the Binomial Expansion
(a+bx)^{n} is never infinite in value, but an infinite expansion might be unless each term is smaller than the last. To prevent this explosion to infinity we can only work with certain values of x. Specifically:
The binomial expansion of (ax+b)^{n} is only valid for |x|<\left|\dfrac{b}{a}\right|
Note: If n is an integer then we do not need to worry about this; we get a finite number of terms in the binomial expansion, so it can never have an infinite value and is thus always valid.
Combinations of Expansions
Sometimes, we will be asked to expand something that contains more than one binomial term. We do this by treating each binomial term individually, then handling them together at the end.
Example: Find the first three terms of the expansion of \color{red}(1+2x)^{-1}\color{blue}(1-3x)^{\frac{1}{2}}
\begin{aligned}\color{red}(1+2x)^{-1}&\color{red}=1-2x+\dfrac{(-1)(-2)}{1\times2}2^{2}x^{2}+...\\[1.2em]&\color{red}=1-2x+4x^{2}+...\end{aligned}
\begin{aligned}\color{blue}(1-3x)^{\frac{1}{2}}&\color{blue}=1+\dfrac{1}{2}(-3)x+\dfrac{\dfrac{1}{2}\times\dfrac{-1}{2}}{1\times2}(-3)^{2}x^{2}+...\\[1.2em]&\color{blue}=1-\dfrac{3}{2}x-\dfrac{\dfrac{1}{4}}{2}\times9x^{2}+...\\[1.2em]&\color{blue}=1-\dfrac{3}{2}x-\dfrac{9}{8}x^{2}+...\end{aligned}
\begin{aligned}&\color{red}(1+2x)^{-1}\color{blue}(1-3x)^{\frac{1}{2}}\color{grey}=\\[1.2em]&\color{red}(1-2x+4x^{2}+...)\color{blue}\left(1-\dfrac{3}{2}x-\dfrac{9}{8}x^{2}+...\right)\\[1.2em]&\color{grey}=\color{red}1\color{blue}\left(1-\dfrac{3}{2}x-\dfrac{9}{8}x^{2}+...\right)\color{grey}-\color{red}2x\color{blue}\left(1-\dfrac{3}{2}x+...\right)\color{grey}\\[1.2em]&+\color{red}4x^{2}\color{blue}(1+...)\\[1.2em]&\color{grey}=1-\dfrac{3}{2}x-\dfrac{9}{8}x^{2}-2x+3x^{2}+4x^{2}+...\\[1.2em]&\color{grey}=1-\dfrac{7}{2}x+\dfrac{47}{8}x^{2}+...\end{aligned}
Infinite Series Binomial Expansions Example Questions
Question 1: Convert these binomials into the form b(1+ax)^{n}.
i) (3+5x)^{3}
ii) (2+x)^{6}
iii) (81+18x)^{\frac{1}{4}}
iv) (3+2x)^{-\frac{1}{2}}
[4 marks]
i) (3+5x)^{3}=3^{3}\left(1+\dfrac{5}{3}x\right)^{3}=27\left(1+\dfrac{5}{3}x\right)^{3}
ii) (2+x)^{6}=2^{6}\left(1+\dfrac{1}{2}x\right)^{6}=64\left(1+\dfrac{1}{2}x\right)^{6}
iii) (81+18x)^{\frac{1}{4}}=81^{\frac{1}{4}}\left(1+\dfrac{18}{81}\right)^{\frac{1}{4}}=3\left(1+\dfrac{2}{9}\right)^{\frac{1}{4}}
iv) (3+2x)^{\frac{1}{2}}=3^{\frac{-1}{2}}\left(1+\dfrac{2}{3}x\right)^{\frac{-1}{2}}=\dfrac{1}{\sqrt{3}}\left(1+\dfrac{2}{3}x\right)^\frac{-1}{2}
Question 2: Find the first four terms of the binomial expansion of (1+4x)^{-2}
[4 marks]</p
\begin{aligned}(1+4x)^{-2}&=1+\big( (-2)\times4x\big) +\left( \dfrac{-2\times(-3)}{1\times2}(4x)^{2}\right) +\left( \dfrac{-2\times(-3)\times(-4)}{1\times2\times3}(4x)^{3}\right) \\[1.2em]&=1-8x+\left( \dfrac{6}{2}\times4^{2}x^{2}\right) -\left( \dfrac{24}{6}\times4^{3}x^{3}\right) \\[1.2em]&=1-8x+\big( 3\times16x^{2}\big) -\big( 4\times64x^{3}\big) \\[1.2em]&=1-8x+48x^{2}-256x^{3}\end{aligned}
Question 3: Find the first three terms of the binomial expansion of (3+x)^{\frac{1}{2}}. For which values of x is this expression valid?
[3 marks]
\begin{aligned}(3+x)^{\frac{1}{2}}&=3^{\frac{1}{2}}\left(1+\dfrac{1}{3}x\right)^{\frac{1}{2}}\\[1.2em]&=\sqrt{3}\left(1+\left( \dfrac{1}{2}\times\dfrac{1}{3}x\right) +\left( \dfrac{\dfrac{1}{2}\times\dfrac{-1}{2}}{1\times2}\times\left(\dfrac{1}{3}x\right)^{2}\right) \right) \\[1.2em]&=\sqrt{3}\left(1+\dfrac{1}{6}x-\left( \dfrac{\left( \dfrac{1}{4}\right) }{2}\times\dfrac{1}{9}x^{2}\right) \right) \\[1.2em]&=\sqrt{3}\left(1+\dfrac{1}{6}x-\left( \dfrac{1}{8}\times\dfrac{1}{9}x^{2}\right) \right) \\[1.2em]&=\sqrt{3}\left(1+\dfrac{1}{6}x-\dfrac{1}{72}x^{2}\right)\end{aligned}
This is valid for |x|<3
Question 4: Find the first three terms of the expansion of \dfrac{(1+2x)^{\frac{1}{2}}}{(3+2x)^{2}}
[6 marks]
Specification Points Covered
D1 – Understand and use the binomial expansion of (a+bx)^n for positive integer n; the notations n! and nCr; link to binomial probabilities
Extend to any rational n, including its use for approximation; be aware that the expansion is valid for \left|\dfrac{bx}{a}\right|<1. (proof not required)
D6 – Use sequences and series in modelling
Related Topics
