AQA A Level Maths: Pure复习笔记4.2.1 General Binomial Expansion

General Binomial Expansion

 

What is the general binomial expansion?

  • The binomial expansion applies for positive integers, n ∈ ℕ
  • The general binomial expansion applies to other types of powers too

4.2.1-GBE-Notes-Diagram-1

  • The general binomial expansion applies for all real numbers, n ∈ℝ
  • Usually fractional and/or negative values of n are used
  • It is derived from (a + b)n, with a = 1 and b = x
  • a = 1 is the main reason the expansion can be reduced so much

4.2.1-GBE-Notes-Diagram-2

  • Unless n ∈ ℕ, the expansion is infinitely long
  • It is only valid for |x| < 1
    • This is another way of writing -1 < x < 1
    • This is often called the validity statement
    • The restriction |x| < 1 means the series will converge
    • Higher powers of x can be ignored (as r → ∞, xr → 0)
    • Only the first few terms of an expansion are needed

     

How do I expand brackets with the general binomial expansion?

STEP 1       Write the expression in the form (1 + x)n
STEP 2       Expand and simplify

Use a line for each term to make things easier to read and follow

Use brackets - fractions and negatives get ugly!

STEP 3       If required, check and state the validity statement

4.2.1-GBE-Notes-Diagram-3

 

How do I use the general binomial expansion when it is (1 + bx)?

STEP 1       Write the expression in the form (1 + bx)n
STEP 2       Replace “x” by “bx” in the expansion

Check carefully to see if b is negative

STEP 3       Expand and simplify

Use a line for each term to make things easier to read and follow

Use brackets

STEP 4       If required, check and state the validity statement
      The validity statement changes   

      Replace “x” with “bx” so now |bx| < 1

4.2.1-GBE-Notes-Diagram-4

4.2.1-GBE-Notes-Diagram-5

Worked Example

4.2.1-GBE-Example-Diagram-14.2.1-GBE-Example-Diagram-2

 

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