Sec. 8.8 Taylor and Maclaurin Series
Just as it is often easier to approximate some numbers (π ≈ 3.14, 2≈ 1.414, etc.), sometimes we want to
approximate a function close to a point with simple terms (i.e., polynomials). The Taylor Series allows us
to do just this.
EX 1 Transparency of Maclaurin series approximations of f(x) = cos x.
Taylor Series: 23
() () ()
()()() () () ()
23! !
nn
fa f a fa
fa faxa xa xa xa
n
′′ ′′′
′
+−+ −+ −++ −+""
Maclaurin Series: 23
(0) (0) (0)
(0) (0) 23! !
nn
ff f
ffx x x x
n
′′ ′′′
′
+
++ +++""
Note the Maclaurin Series is a Taylor Series centered about 0.
Note that the
(
)
()
!
n
n
f
a
cn
= is the coefficient of each term with 0! =1 and
(
)
0
f
f=.
EX 2 Find the Maclaurin Series for ( ) 2
x
fx
=
using the definition of the Maclaurin series. Also find the
radius of convergence.
