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Taylor's Inequality

In a Nut Shell: Taylor series can be used to approximate a given function. The accuracy

depends on the number of terms taken. Taylor's Inequality provides a way to estimate the

accuracy of a Taylor's series expansion of a function.

Recall the Taylor series expansion of f(x) is given by:

∞

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.

Taylor's Inequality: If | f⁽ⁿ⁺¹⁾ (x) | ≤ M for | x ˗ a | ≤ d, then the remainder

R_n(x) of the Taylor series satisfies the inequality

| R_n(x) | ≤ M / ( n + 1)! | x ˗ a | ^{n +
1} for | x ˗ a | ≤ d

where | R_n(x) | is the magnitude of the remainder using n terms

M is the largest value of | f⁽ⁿ⁺¹⁾(x) | in the interval a ˗ d ≤ x ≤ a + d

( n + 1 )! is the factorial function

| x ˗ a | is the size of the interval over which f(x) is represented

n + 1 is the exponent

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