8.5. TAYLOR POLYNOMIALS AND TAYLOR SERIES
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8.5 Taylor Polynomials and Taylor Series
Motivating Questions
In this section, we strive to understand the ideas generated by the following important
questions:
• What is a Taylor polynomial? For what purposes are Taylor polynomials used?
• What is a Taylor series?
• How are Taylor polynomials and Taylor series different? How are they related?
• How do we determine the accuracy when we use a Taylor polynomial to approximate a function?
Introduction
In our work to date in Chapter 8, essentially every sum we have considered has been a
sum of numbers. In particular, each infinite series that we have discussed has been a series
of real numbers, such as
1 +
1
2
+
1
4
+ · · · +
1
2 k + · · · =
∞
k=0
1
2 k .
(8.18)
In the remainder of this chapter, we will expand our notion of series to include series that
involve a variable, say x. For instance, if in the geometric series in Equation (8.18) we
replace the ratio r =
1
2 with the variable x, then we have the infinite (still geometric) series
1 + x + x
2 + · · · + x
k + · · · =
∞
k=0
x
k .
(8.19)
Here we see something very interesting: since a geometric series converges whenever its
ratio r satisfies |r | < 1, and the sum of a convergent geometric series is
a
1−r , we can say
that for |x| < 1,
1 + x + x
2 + · · · + x
k + · · · =
1
1 − x
.
(8.20)
Note well what Equation (8.20) states: the non-polynomial function
1
1−x on the right is
equal to the infinite polynomial expresssion on the left. Moreover, it appears natural to
truncate the infinite sum on the left (whose terms get very small as k gets large) and say,
for example, that
1 + x + x
2 + x
3 ≈
1
1 − x