BINOMIAL FOR NEGATIVE OR FRACTION n (SCHOOL)

BINOMIAL FOR NEGATIVE OR FRACTION n (SCHOOL)

What is it?

For |x|<1, (1+x)ⁿ = 1 + n x + n(n-1)/2! x² + n(n-1)n-23! x^3 + ... even if n is not a positive integer. You usually keep the first few terms (approximation). (1-x)^{-1} = 1+x+x^2+... . Meaning: the same pattern of falling factors, but it does not stop. Only valid as an infinite series in a disc |x|<1 in school wording.

In simple words

• For |x|<1, (1+x)ⁿ = 1 + n x + n(n-1)/2!
• x² + n(n-1)n-23!
• x^3 + ...
• Infinite binomial needs |x|<1 in this course. Positive integer n does not need that.

Important terms

BINOMIAL FOR NEGATIVE OR FRACTION n (SCHOOL): For |x|<1, (1+x)^n = 1 + n x + n(n-1)/2! x^2 + n(n-1)n-23! x^3 + ... even if n is not a positive integer. You usually keep the first few terms (approximation). (1-x)^{-1} = 1+x+x^2+... . Meaning: the same pattern of falling factors, but it does not stop. Only valid as an infinite series in a disc |x|<1 in school wording.

1. FIRST TERMS: (1+x)12} ~ 1 + (12)x - (18)x^2 + ... . Compute n(n-1)/2 for the x^2 coeff. Stop at the term the question asks.

2. USES: Approximate sqrt(1.02) as (1+0.02)12} with two terms. (1-x)^-1 geometric. Do not plug |x|>1 into the infinite expansion as if it were exact.

What you should learn

• State the main ideas and notation used in binomial for negative or fraction n (school).
• Apply the rules of binomial for negative or fraction n (school) in a clear, step-by-step solution.
• Check signs, conditions, and the final answer for mathematical accuracy.

Examples

• n=-1: all +1 coeffs if (1-x)^-1
• n=12 square root
• Show 2! 3! in the bottoms

FIRST TERMS

(1+x)12} ~ 1 + (12)x - (18)x^2 + ... . Compute n(n-1)/2 for the x^2 coeff. Stop at the term the question asks.

USES

Approximate sqrt(1.02) as (1+0.02)12} with two terms. (1-x)^-1 geometric. Do not plug |x|>1 into the infinite expansion as if it were exact.

Full school-level explanation

For |x|<1, (1+x)ⁿ = 1 + n x + n(n-1)/2! x² + n(n-1)n-23! x^3 + ... even if n is not a positive integer. You usually keep the first few terms (approximation). (1-x)^{-1} = 1+x+x^2+... . Meaning: the same pattern of falling factors, but it does not stop. Only valid as an infinite series in a disc |x|<1 in school wording.

Write three terms, then a named series:

WHAT — BINOMIAL FOR NEGATIVE OR FRACTION n (SCHOOL)

For |x|<1, (1+x)ⁿ = 1 + n x + n(n-1)/2! x² + n(n-1)n-23! x^3 + ... even if n is not a positive integer. You usually keep the first few terms (approximation). (1-x)^{-1} = 1+x+x^2+... . Meaning: the same pattern of falling factors, but it does not stop. Only valid as an infinite series in a disc |x|<1 in school wording.

RULE — 1. FIRST TERMS

(1+x)12} ~ 1 + (12)x - (18)x^2 + ... . Compute n(n-1)/2 for the x^2 coeff. Stop at the term the question asks.

METHOD — 2. USES

Approximate sqrt(1.02) as (1+0.02)12} with two terms. (1-x)^-1 geometric. Do not plug |x|>1 into the infinite expansion as if it were exact.

WORKED EXAMPLE

Worked: first three terms of (1+x)^-1. n=-1, 1 + (-1)x + (-1)-22 x^2 + ... = 1 - x + x^2 - ... matches 11+x geometric with ratio -x if |x|<1

FIRST TERMS

(1+x)12} ~ 1 + (12)x - (18)x^2 + ... . Compute n(n-1)/2 for the x^2 coeff. Stop at the term the question asks.

Remember

• stop where asked

Examples

• n=-1: all +1 coeffs if (1-x)^-1
• n=12 square root
• Show 2! 3! in the bottoms

USES

Approximate sqrt(1.02) as (1+0.02)12} with two terms. (1-x)^-1 geometric. Do not plug |x|>1 into the infinite expansion as if it were exact.

Remember

• small x

Examples

• sqrt(1+x) approx 1+x/2
• Need |x|<1
• Finite n still uses the finite theorem

How to answer a binomial for negative or fraction n (school) question

How it works

• Identify the given quantities, notation, and required result.
• Choose the matching rule and write it before substitution.
• Work one algebraic step at a time, keeping brackets and signs clear.
• Check restrictions, simplify the result, and verify it where possible.

Remember

• falling product / r!
• |x|<1 for the infinite story
• 1 - x + x^2
• |x|<1 if infinite
• stop where asked
• small x
• falling product / r!
• |x|<1 for the infinite story

How the ideas connect

Worked: first three terms of (1+x)^-1. n=-1, 1 + (-1)x + (-1)-22 x^2 + ... = 1 - x + x^2 - ... matches 11+x geometric with ratio -x if |x|<1

Deeper understanding

Go beyond memorising BINOMIAL FOR NEGATIVE OR FRACTION n (SCHOOL). Connect the definition to reasons, relationships, exceptions, and exam wording.

How the ideas connect

Worked: first three terms of (1+x)^-1. n=-1, 1 + (-1)x + (-1)-22 x^2 + ... = 1 - x + x^2 - ... matches 11+x geometric with ratio -x if |x|<1

Common mistakes

Exam focus

Core Idea

1. FIRST TERMS

2. USES

Worked Example

Exam focus

Quick revision

• falling product / r!
• |x|<1 for the infinite story
• 1 - x + x^2
• |x|<1 if infinite
• BINOMIAL FOR NEGATIVE OR FRACTION n (SCHOOL): For |x|<1, (1+x)^n = 1 + n x + n(n-1)/2! x^2 + n(n-1)n-23! x^3 + ... even if n is not a positive
• 1. FIRST TERMS: (1+x)12} ~ 1 + (12)x - (18)x^2 + ... . Compute n(n-1)/2 for the x^2 coeff. Stop at the term the

Infinite binomial needs |x|<1 in this course. Positive integer n does not need that.

For long answers on BINOMIAL FOR NEGATIVE OR FRACTION n (SCHOOL): define clearly → explain the main parts → give one example → state one exam caution.