Maclaurin series

Maclaurin Series Basics

  • Maclaurin series are a type of power series that provide approximations to various functions.
  • They are a special case of the Taylor series expansion, centred at zero.
  • Whereas general Taylor series can be centred on any value of x (which we call a), Maclaurin series are centred at x = 0 (a = 0), meaning they approximate the function near x = 0.
  • The formal definition is f(x) = f(0) + f'(0)x + f''(0)x^2/2! + f'''(0)x^3/3! + ...
  • This shows that each term of the series involves a derivative of the function, evaluated at zero, divided by the factorial of the power of x.

Important Examples of Maclaurin Series

  • The Maclaurin series for e^x is 1 + x + x^2/2! + x^3/3! + ...
  • The Maclaurin series for sin(x) is x - x^3/3! + x^5/5! - x^7/7! + ...
  • The Maclaurin series for cos(x) is 1 - x^2/2! + x^4/4! - x^6/6! + ...
  • These functions can be approximated by cutting the series off after a certain number of terms. This is called a Maclaurin polynomial.

Working with Maclaurin Series

  • To generate the series, one must be able to differentiate the function multiple times and substitute x = 0 into the function and its derivatives.
  • Functions that can’t be differentiated aren’t suited to Maclaurin series representation.
  • Knowledge and use of the chain rule, product rule, and quotient rule are often necessary for deriving these series.
  • The behaviour of a function for large x can’t be inferred from its Maclaurin series; this method only gives an approximation near x = 0.
  • When comparing to actual function values, it’s important to consider the radius of convergence and the degree of accuracy required.

Applications of Maclaurin Series

  • Maclaurin series have applications in physics, engineering, computer science, and other fields.
  • They help in solving differential equations by turning them into infinite series.
  • In maths, they’re used to find limits and to compute function values that can’t be computed directly.
  • They also allow for analytic continuation of functions.