# 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.