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 calla), Maclaurin series are centred atx = 0(a = 0), meaning they approximate the function nearx = 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^xis1 + x + x^2/2! + x^3/3! + ... - The Maclaurin series for
sin(x)isx - x^3/3! + x^5/5! - x^7/7! + ... - The Maclaurin series for
cos(x)is1 - 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 = 0into 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
xcan’t be inferred from its Maclaurin series; this method only gives an approximation nearx = 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.