Partial fractions with binomial series | CAIE 9709 Pure Mathematics 3 Notes

Pure Mathematics 3

Mathematics9709

Split first, then expand each piece

Worked example

One fraction becomes two familiar series

\frac1{(1-x)(2+x)}

=\frac1{3(1-x)}+\frac1{3(2+x)}

\frac1{3(1-x)}

=\frac13(1+x+x^2+x^3+\cdots)

\frac1{3(2+x)}

\begin{aligned}&=\frac16\left(1-\frac{x}{2}+\frac{x^2}{4}\right.\\&\left.\qquad-\frac{x^3}{8}+\cdots\right)\end{aligned}

Add like powers from the two series.

Answer

\frac12+\frac14x

+\frac38x^2+\frac5{16}x^3+\cdots

Common mistake

Do not combine the constants and powers before applying the outside partial-fraction coefficients.

Intersect every validity condition

Keep only the shared interval

first piece: −1 < x < 1

second piece: −2 < x < 2

intersection: −1 < x < 1

A combined series is valid only where every component series is valid.

Key idea

The narrowest valid interval controls the combined expansion.

Keep decomposition, coefficients and range visible

Your turnIndependent check [6 marks]

Expand $\dfrac1{(1-2x)(1+x)}$ up to the term in $x^2$ and state the range of validity.

Show worked answer

\frac1{(1-2x)(1+x)}

=\frac{2}{3(1-2x)}+\frac1{3(1+x)}

\begin{aligned}&=\frac23(1+2x+4x^2)\\&\quad+\frac13(1-x+x^2)+\cdots\end{aligned}

Answer

1+x+3x^2+\cdots

The two conditions are $|2x|<1$ and $|x|<1$ .

Answer

-\frac12<x<\frac12

Split first, then expand each piece

Worked example

One fraction becomes two familiar series

\frac1{(1-x)(2+x)}

=\frac1{3(1-x)}+\frac1{3(2+x)}

\frac1{3(1-x)}

=\frac13(1+x+x^2+x^3+\cdots)

\frac1{3(2+x)}

\begin{aligned}&=\frac16\left(1-\frac{x}{2}+\frac{x^2}{4}\right.\\&\left.\qquad-\frac{x^3}{8}+\cdots\right)\end{aligned}

Add like powers from the two series.

Answer

\frac12+\frac14x

+\frac38x^2+\frac5{16}x^3+\cdots

Common mistake

Do not combine the constants and powers before applying the outside partial-fraction coefficients.

Intersect every validity condition

Keep only the shared interval

first piece: −1 < x < 1

second piece: −2 < x < 2

intersection: −1 < x < 1

A combined series is valid only where every component series is valid.

Key idea

The narrowest valid interval controls the combined expansion.

Keep decomposition, coefficients and range visible

Your turnIndependent check [6 marks]

Expand $\dfrac1{(1-2x)(1+x)}$ up to the term in $x^2$ and state the range of validity.

Show worked answer

\frac1{(1-2x)(1+x)}

=\frac{2}{3(1-2x)}+\frac1{3(1+x)}

\begin{aligned}&=\frac23(1+2x+4x^2)\\&\quad+\frac13(1-x+x^2)+\cdots\end{aligned}

Answer

1+x+3x^2+\cdots

The two conditions are $|2x|<1$ and $|x|<1$ .

Answer

-\frac12<x<\frac12