Taylor and Maclaurin series

Taylor and Maclaurin series write a function as an infinite polynomial. A few terms give an excellent approximation near the centre, which is how calculators work out things like sin and eˣ.

Key facts

  • Maclaurin: f(x) = f(0) + f′(0)x + f″(0)x²/2! + …
  • eˣ = 1 + x + x²/2! + x³/3! + …
  • sin x = x − x³/3! + x⁵/5! − …; cos x = 1 − x²/2! + x⁴/4! − …
  • ln(1 + x) = x − x²/2 + x³/3 − … for −1 < x ≤ 1.

Worked examples

Example 1

Use the first three terms of the series for eˣ to estimate e^0.1.

  1. 1 + 0.1 + 0.01/2 = 1.105
  2. The true value is 1.10517…

Answer: 1.105

Example 2

Find the Maclaurin series of cos 2x up to x⁴.

  1. 1 − (2x)²/2 + (2x)⁴/24
  2. = 1 − 2x² + 2x⁴/3

Answer: 1 − 2x² + 2x⁴/3