CIE A Level Maths: Pure 3复习笔记1.5.4 Approximating values

What is the general binomial expansion?

• The general binomial expansion, as given in the formula booklet, is

• If n ∈ ℕ then the expansion is finite (see Binomial Expansion)
• Otherwise the expansion is infinitely long
  • It is only valid for |x| < 1 (-1 < x < 1)
  • Only the first few terms of an expansion are usually needed

How do I use a binomial expansion to approximate a value?

•  Ignoring higher powers of x leads to an approximation
• The more terms the closer the approximation is to the true value
• For most purposes, squared or cubed terms are accurate enough

STEP 1  Compare the value you are approximating to (a + bx)ⁿ
STEP 2  Solve the appropriate equation to find the value of x
STEP 3  Substitute this value of x into the expansion to find the approximation