Adapting a binomial series and its validity | CAIE 9709 Pure Mathematics 3 Notes

Chapters

Pure Mathematics 3

Mathematics9709

Normalize the bracket before expanding

Normalize before checking validity

start

(a + bx)ⁿ

→

normalize

aⁿ(1 + u)ⁿ

new inside

u = bx/a

validity gate

|u| < 1 → solve |bx/a| < 1

Keep the outside factor aⁿ on every term.

Validity belongs to the normalized inside u, not automatically to x.

(a+bx)^n=a^n\left(1+\frac{b}{a}x\right)^n

The series variable is $u=bx/a$ , not automatically $x$ .

Examiner note

A correct expansion without its requested range of validity is incomplete.

Carry the outside factor and solve the absolute inequality

Worked example

Expand an adapted negative power

(2-3x)^{-2}=\frac14\left(1-\frac32x\right)^{-2}

=\frac14\left(1+3x+\frac{27}{4}x^2+\cdots\right)

Answer

\frac14+\frac34x+\frac{27}{16}x^2+\cdots

The series is valid when:

\left|-\frac32x\right|<1

Answer

-\frac23<x<\frac23

Common mistake

The validity statement is not

x<1

. It comes from the magnitude of the complete substituted inside.

Expansion and validity are one linked answer

Your turnIndependent check [5 marks]

Expand $(4+x)^{-1/2}$ up to the term in $x^2$ and state the range of validity.

Show worked answer

(4+x)^{-1/2}=\frac12\left(1+\frac{x}{4}\right)^{-1/2}

Answer

\frac12-\frac{x}{16}+\frac{3x^2}{256}+\cdots

\left|\frac{x}{4}\right|<1

Answer

-4<x<4

Normalize the bracket before expanding

Normalize before checking validity

start

(a + bx)ⁿ

→

normalize

aⁿ(1 + u)ⁿ

new inside

u = bx/a

validity gate

|u| < 1 → solve |bx/a| < 1

Keep the outside factor aⁿ on every term.

Validity belongs to the normalized inside u, not automatically to x.

(a+bx)^n=a^n\left(1+\frac{b}{a}x\right)^n

The series variable is $u=bx/a$ , not automatically $x$ .

Examiner note

A correct expansion without its requested range of validity is incomplete.

Carry the outside factor and solve the absolute inequality

Worked example

Expand an adapted negative power

(2-3x)^{-2}=\frac14\left(1-\frac32x\right)^{-2}

=\frac14\left(1+3x+\frac{27}{4}x^2+\cdots\right)

Answer

\frac14+\frac34x+\frac{27}{16}x^2+\cdots

The series is valid when:

\left|-\frac32x\right|<1

Answer

-\frac23<x<\frac23

Common mistake

The validity statement is not

x<1

. It comes from the magnitude of the complete substituted inside.

Expansion and validity are one linked answer

Your turnIndependent check [5 marks]

Expand $(4+x)^{-1/2}$ up to the term in $x^2$ and state the range of validity.

Show worked answer

(4+x)^{-1/2}=\frac12\left(1+\frac{x}{4}\right)^{-1/2}

Answer

\frac12-\frac{x}{16}+\frac{3x^2}{256}+\cdots

\left|\frac{x}{4}\right|<1

Answer

-4<x<4