Barrons AP Calculus

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.

Maclaurin   Series

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.

SOLUTION:

Thus,