Barrons AP Calculus - David Bock

Since when x = 0 we see that c = 1, we have

Note that this is a geometric series with ratio r = x and with a = 1; if |x| < 1, its sum is

C3. Finding a Power Series for a Function: Taylor and Maclaurin

Series.

If a function f (x) is representable by a power series of the form

c 0 + c 1 (x − a) + c 2 (x − a)^2 + ··· + cn (x − a)n +···

on an interval |x − a| < r, then the coefficients are given by

Taylor series

The series

is called the Taylor series of the function f about the number a. There is never more than one power
series in (x − a) for f (x). It is required that the function and all its derivatives exist at x = a if the
function f (x) is to generate a Taylor series expansion.
When a = 0 we have the special series

called the Maclaurin series of the function f; this is the expansion of f about x = 0.

EXAMPLE 40

Find the Maclaurin series for f (x) = ex.

SOLUTION: Here f ′(x) = ex, ..., f (n) (x) = ex, ..., for all n. Then

f ′(0) = 1, ..., f (n) (0) = 1, ...,

for all n, making the coefficients

EXAMPLE 41

Find the Maclaurin expansion for f (x) = sin x.