The binomial series (1 + x)^n

Q: Do I need to memorize the binomial series formula?

No, the full series for (1 + x)^n is provided on the 4PM1 formula sheet, but you must remember the validity condition |x| < 1 for rational (non-positive integer) n.

Q: When is the binomial expansion of (1 + x)^n finite?

When n is a positive integer, the expansion terminates after n+1 terms, with no validity restriction on x.

Q: How do I expand (a + bx)^n using the standard (1 + x)^n formula?

Factor out a from the bracket first: , then apply the standard binomial series to the term.

Edexcel International GCSE Further Pure Mathematics· Section S6 (spec 6A)· 12 min read

1. 1. Finite Expansion: n is a Positive Integer★★☆☆☆⏱ 3 min

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When n is a positive integer, the binomial series for $(1 + x)^{n}$ produces a finite, terminating expansion with exactly n+1 terms. There is no validity restriction on x for this case, as the series stops after the $x^{n}$ term.

📘 Definition

Terminating binomial expansion

A finite expansion of $(1 + x)^{n}$ for positive integer n, with no infinite terms, valid for all real values of x.

📐 Worked Example

Fully expand $(1 + 2 x)^{4}$ .

1
Recall the binomial series for positive integer n, matching the formula sheet:
2
$(1 + x)^n = 1 + nx + \frac{n(n-1)}{2!}x^2 + ... + x^n$
3
Substitute $n = 4$ and $x = 2 x$ into the formula:
4
$1 + 4(2x) + \frac{4 \times 3}{2!}(2x)^2 + \frac{4 \times 3 \times 2}{3!}(2x)^3 + \frac{4!}{4!}(2x)^4$
5
Simplify each term to get the final expansion:
6
$1 + 8x + 24x^2 + 32x^3 + 16x^4$

Exam tip:

For positive integer n, you can use either combination notation $(r n)$ or the general term formula from the formula sheet, both produce identical results.

2. 2. Infinite Expansion: n is Rational, |x| < 1★★★☆☆⏱ 4 min

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When n is a negative integer, fraction, or any rational number that is not a positive integer, the binomial series produces an infinite expansion. This expansion is only mathematically valid when $∣ x ∣ < 1$ : you must state this condition in every exam answer for this case.

📘 Definition

Validity condition (rational n)

The infinite expansion of $(1 + x)^{n}$ for $n \in Q$ (non-positive integer) is only convergent when the absolute value of x is less than 1, written $∣ x ∣ < 1$ .

📐 Worked Example

Find the first three non-zero terms of the expansion of $(1 - x)^{- 1/2}$ , and state the range of x for which the expansion is valid.

1
Rewrite the expression as $(1 + (- x))^{n}$ where $n = - 1/2$ .
2
Apply the binomial series formula from the formula sheet:
3
$1 + nx + \frac{n(n-1)}{2!}x^2 + ...$
4
Substitute $n = - 1/2$ and replace x with $- x$ :
5
$1 + (-1/2)(-x) + \frac{(-1/2)(-1/2 - 1)}{2!}(-x)^2 + ...$
6
Simplify each term:
7
$1 + \frac{1}{2}x + \frac{3}{8}x^2 + ...$
8
State the validity condition: $∣ - x ∣ < 1 ⟹ ∣ x ∣ < 1$

Exam tip:

Always substitute the entire argument of the bracket (e.g., $- x$ for $(1 - x)^{n}$ ) into the standard formula, not just the coefficient of x.

3. 3. Expanding $(a + bx)^n$ by Factoring Out a★★★★☆⏱ 3 min

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To apply the standard $(1 + x)^{n}$ series to expressions of the form $(a + b x)^{n}$ , first factor out the constant a from the bracket to rewrite the expression in the standard form. Adjust the validity condition to match the new argument of the $(1 + k x)$ term.

📐 Worked Example

Find the first three terms of the expansion of $(4 + x)^{- 1}$ , and state the validity condition.

1
Factor out 4 from the bracket, applying the exponent -1 to the factored constant:
2
$(4(1 + x/4))^{-1} = 4^{-1} \left(1 + \frac{x}{4}\right)^{-1} = \frac{1}{4}\left(1 + \frac{x}{4}\right)^{-1}$
3
Expand $(1 + \frac{x}{4})^{- 1}$ using the binomial series:
4
$1 + (-1)\left(\frac{x}{4}\right) + \frac{(-1)(-2)}{2!}\left(\frac{x}{4}\right)^2 + ... = 1 - \frac{x}{4} + \frac{x^2}{16} + ...$
5
Multiply through by $1/4$ to get the final expansion:
6
$\frac{1}{4} - \frac{x}{16} + \frac{x^2}{64} + ...$
7
State the validity condition: $\frac{x}{4} < 1 ⟹ ∣ x ∣ < 4$

Exam tip:

Never forget to apply the exponent n to the factored out constant term: $(a \times f (x))^{n} = a^{n} [f (x)]^{n}$ , not $a [f (x)]^{n}$ — this is one of the most common mark-losing errors for this topic.

4. 4. Applications: Terms, Coefficients and Approximations★★★★☆⏱ 2 min

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Exam questions frequently ask you to find specific terms (e.g., the coefficient of $x^{3}$ ) or use the first 2-3 terms of an expansion to approximate values of expressions for small x, where the higher-order terms are negligible.

📐 Worked Example

Use the first three terms of the expansion of $1 + 0.04$ to find an approximate value of $1.04$ , giving your answer to 3 decimal places.

1
Rewrite the expression as $(1 + 0.04)^{1/2}$ .
2
Calculate the first three terms of the expansion:
3
$1 + \frac{1}{2}(0.04) + \frac{(1/2)(-1/2)}{2!}(0.04)^2$
4
Simplify each term:
5
$1 + 0.02 - 0.0002 = 1.0198$
6
Round to 3 decimal places to get the final approximate value: 1.020

5. Common Pitfalls

Wrong move:

Forgetting to state the validity condition $∣ x ∣ < 1$ for rational n expansions

Why:

Examiners explicitly award marks for stating this condition, so omitting it loses easy, guaranteed marks

Correct move:

Always write the validity condition immediately after any infinite binomial expansion answer, adjusting for the argument of the bracket if needed

Wrong move:

Failing to apply the exponent n to the factored out constant when expanding $(a + b x)^{n}$

Why:

This leads to all terms in the expansion being scaled incorrectly, resulting in full loss of marks for the expansion

Correct move:

When factoring a out of $(a + b x)^{n}$ , write it as $a^{n} (1 + \frac{b}{a} x)^{n}$ , not $a (1 + \frac{b}{a} x)^{n}$

Wrong move:

Using the validity condition $∣ x ∣ < 1$ for positive integer n expansions

Why:

Finite expansions for positive integer n are valid for all x, so this condition is unnecessary and may cost you a mark for incorrect validity

Correct move:

Only state validity conditions for expansions where n is not a positive integer

Wrong move:

Substituting $3 x$ instead of $- 3 x$ into the standard formula for $(1 - 3 x)^{n}$

Why:

This leads to incorrect signs for all odd-powered terms, losing marks even if your expansion method is otherwise correct

Correct move:

Rewrite the bracket as $(1 + (- 3 x))^{n}$ before substituting into the standard $(1 + x)^{n}$ formula

6. Quick Reference Cheatsheet

Case	Expansion Form	Validity	General r-th Term
n positive integer	$(1 + x)^{n} = 1 + n x + \frac{n ( n - 1 )}{2 !} x^{2} + ... + x^{n}$	All real x	$(r n) x^{r}$
n rational (non-positive integer)	$(1 + x)^{n} = 1 + n x + \frac{n ( n - 1 )}{2 !} x^{2} + ...$	$∣ x ∣ < 1$	$\frac{n ( n - 1 ) ... ( n - r + 1 )}{r !} x^{r}$
$(a + b x)^{n}$ (non-positive integer n)	$a^{n} (1 + \frac{b}{a} x)^{n}$	$∣ x ∣ < \frac{a}{b}$	$a^{n} \times \frac{n ( n - 1 ) ... ( n - r + 1 )}{r !} (\frac{b}{a} x)^{r}$

7. Frequently Asked

Do I need to memorize the binomial series formula?

No, the full series for (1 + x)^n is provided on the 4PM1 formula sheet, but you must remember the validity condition |x| < 1 for rational (non-positive integer) n.

When is the binomial expansion of (1 + x)^n finite?

When n is a positive integer, the expansion terminates after n+1 terms, with no validity restriction on x.

How do I expand (a + bx)^n using the standard (1 + x)^n formula?

Factor out a from the bracket first: $(a + b x)^{n} = a^{n} (1 + \frac{b}{a} x)^{n}$ , then apply the standard binomial series to the $(1 + \frac{b}{a} x)$ term.

Going deeper

official resourceEdexcel 4PM1 SpecificationOfficial specification for full topic guidance

What's Next

Now that you have mastered the binomial series for $(1 + x)^{n}$ , you can apply this skill to solve more complex algebra problems in your Edexcel IGCSE Further Pure Math exam. This topic is frequently combined with algebraic simplification and coordinate geometry questions, so make sure you practice connecting these skills. The next topics in the S6 unit build directly on this foundation, so you should move on to practicing past paper questions focused on binomial series applications to solidify your understanding before moving on to differentiation in S7.