Binomial expansion for rational powersEdexcel A-Level Maths: Revision notes

For any rational $n$ (negative or fractional as well as positive integer), $(1+x)^n=1+nx+\frac{n(n-1)}{2!}x^2+\frac{n(n-1)(n-2)}{3!}x^3+\ldots,\qquad |x|<1.$ When $n$ is a positive integer the series stops and is valid for all $x$. For any other rational $n$ it never stops, and it is valid only when $|x|<1$. Example: $(1+4x)^{-\frac12}=1+\left(-\frac12\right)(4x)+\frac{\left(-\frac12\right)\left(-\frac32\right)}{2!}(4x)^2+\ldots=1-2x+6x^2-\ldots$ Work term by term: the numerator multiplies $n$ by $n-1$, $n-2$, ..., and the denominator is the matching factorial. Use brackets for every negative or fractional number.

To expand $(a+bx)^n$ with $a\ne1$, take out $a^n$ first: $(a+bx)^n=a^n\left(1+\frac bax\right)^n.$ Then expand $\left(1+\frac bax\right)^n$ using the formula with $u=\frac bax$. Example: $\sqrt{4+x}=2\left(1+\frac x4\right)^{\frac12}=2\left[1+\frac x8-\frac{x^2}{128}+\ldots\right]=2+\frac x4-\frac{x^2}{64}+\ldots$ Example: $\frac{1}{\sqrt{9-2x}}=\frac13\left(1-\frac{2x}{9}\right)^{-\frac12}=\frac13+\frac{x}{27}+\frac{x^2}{162}+\ldots$ The factor $a^n$ must be applied to every term: $4^{\frac12}=2$, $9^{-\frac12}=\frac13$.

The expansion of $(1+u)^n$ is valid only if $|u|<1$. For $(a+bx)^n$ this means $\left|\frac{bx}{a}\right|<1\quad\Longleftrightarrow\quad |x|<\left|\frac ab\right|.$ For $(1+4x)^{-\frac12}$: $|4x|<1$, so $|x|<\frac14$. For $\sqrt{4+x}$: $\left|\frac x4\right|<1$, so $|x|<4$. For $(9-2x)^{-\frac12}$: $\left|\frac{2x}{9}\right|<1$, so $|x|<\frac92$. The range depends on the bracket, not on the power $n$. A value of $x$ outside the range gives nonsense even if the sum is calculable, so check that $x$ is inside the range before using the series.

To approximate a number, choose $x$ so that the expression equals the number, check it is within the range of validity, and substitute into the first few terms. To estimate $\frac{1}{\sqrt{8.8}}$ use $h(x)=\frac{1}{\sqrt{9-2x}}$ at $x=0.1$ (since $9-0.2=8.8$): $\frac13+\frac{0.1}{27}+\frac{0.01}{162}=0.3371$. A small $x$ gives a better approximation, and more terms improve it further. The percentage error is $\frac{|\text{estimate}-\text{exact}|}{\text{exact}}\times100$. Calculate the exact value on your calculator to measure the error.

A rational function such as $\frac{5-x}{(1+x)(1-2x)}$ can be expanded by first splitting into partial fractions: $\frac{2}{1+x}+\frac{3}{1-2x}$. Then expand each part: $2(1+x)^{-1}=2-2x+2x^2-\ldots,\qquad 3(1-2x)^{-1}=3+6x+12x^2+\ldots$ and add: $5+4x+14x^2+\ldots$ Each part has its own range of validity: $|x|<1$ and $|2x|<1$. The combined expansion is valid only where both hold, so $|x|<\frac12$, the more restrictive range. Always comment on the range when asked, and check that any value you substitute lies inside it.