Taylor’s formula

Suppose we’re working with a function f(x) that is continuous and has n+1 continuous derivatives on an interval about x = 0. We can approximate f near 0 by a polynomial of degree n:

• For n = 0, the best constant approximation near 0 is which matches f at 0.

• For n = 1, the best linear approximation near 0 is . Note that matches f at 0 and matches at 0.

• For n = 2, the best quadratic approximation near 0 is . Note that , , and match , , and , respectively, at 0.

Continuing this process,

.

This is the Taylor polynomial of degree n about 0 (also called the Maclaurin series of degree n). More generally, if f has n+1 continuous derivatives at x = a, the Taylor series of degree n about a is

.

This formula approximates f (x) near a. Taylor’s Theorem gives bounds for the error in this approximation:

Taylor’s Theorem:

Suppose f has n+1 continuous derivatives on an open interval containing a. Then for each x in the interval,

,

where the error term satisfies for some c between a and x.

This form for the error , derived in 1797 by Joseph Lagrange, is called the Lagrange formula for the remainder. The infinite Taylor series converges to f,

,

if and only if .

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