# 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.