Math Help

The Taylor Series and Maclaurin Series Expansion of Functions, f(x)

In a Nut Shell: Sometimes it’s helpful to represent functions, f(x), as a power series

(i.e. polynomials). These series expansions then can be used to evaluate difficult limits

and integrals of complicated functions that cannot be evaluated by standard methods.

The Taylor series expansion of f(x)

∞

f(x) = ∑ [(f⁽ⁿ⁾(a)/n!] (x – a)ⁿ= f(a) + f’(a) (x – a) + [f’’(a)/2!] (x-a)² + . . .

n = 0

where f(x) is any function, f⁽ⁿ⁾(a) is the nth derivative of f(x) at x = a,

n! is the factorial function i.e. 3! = 3(2)(1)

The series is centered about x = a.

The Maclaurin Series expansion of f(x) is the same as the Taylor Series

expansion except it is centered at a = 0

∞

f(x) = ∑ [(f⁽ⁿ⁾(0)/n!] (x )ⁿ= f(0) + f’(0) (x) + [f’’(0)/2!] (x)² + . . .

n = 0

Example of Taylor series expansion of f(x) = e^xabout a = 1

f(x) = f’(x) = f’’(x) = . . . = e^x

f(1) = f’(1) = f’’(1) = . . . = e

So f(x) = e + e (x-1) + [e/2!](x – 1)²+ [e/3!](x – 1)³+ [e/4!](x – 1)⁴+ . . .

Or f(x) = e^x=e { 1 + (x - 1) + [1/2](x – 1)²+ [1/3!](x – 1)³+ [1/4!](x – 1)⁴+ . . . }

Example of Maclaurin series expansion of f(x) = e^x(about x = 0)

f(x) = f’(x) = f’’(x) = . . . = e^x

f(0) = f’(0) = f’’(0) = . . . = 1

So f(x) = e^x = 1 + x + x²/2!+ x³/3!+ x⁴/4! + . . .

Return to Calculus 2 Notes Involving Sequences, Infinite Series, etc