Math 152 – Spring 2016
Section 10.7
1 of 8
Section 10.7 – Taylor and Maclaurin Series
Question. How do we rewrite general functions as power series?
Suppose we have a function f (x) that is written as the following series
f (x) = c0 + c1 (x − a) + c2 (x − a)2 + c3 (x − a)3 + c4 (x − a)4 · · ·
that converges when |x − a| < R.
f 0 (x) =
f 0 (a) =
f 00 (x) =
f 00 (a) =
f 000 (x) =
f 000 (a) =
In general, we have the formula
f (n) (a) =
and solving this for cn we have
cn =
Theorem. If f has a power series representation (expansion) at a, that is if
f (x) =
∞
X
cn (x − a)n
|x − a| < R
n=0
then its coefficients are given by the formula
cn =
f (n) (a)
n!
Note. We use the convention that f (0) (x) is the original function f (x) and 0! = 1.
Definition. Substituting the formula for cn into the power series gives what is called
the Taylor series of the function f at a or about a or centered at a:
f (x) =
∞
X
f (n) (a)
(x − a)n
n!
n=0
= f (a) + f 0 (a)(x − a) +
f 00 (a)
f 000 (a)
f (4) (a)
(x − a)2 +
(x − a)3 +
(x − a)4 + · · ·
2!
3!
4!
Math 152 – Spring 2016
Section 10.7
2 of 8
The special case when a = 0 is called a Maclaurin Series:
f (x) =
∞
X
f (n) (0) n
x
n!
n=0
= f (0) + f 0 (0)x +
f 00 (0) 2 f 000 (0) 3 f (4) (0) 4
x +
x +
x + ···
2!
3!
4!
Note.
1. If f has a power series expansion at a, then this power series must be the
Taylor Series centered at a. This means that f is equal to the sum of its Taylor
Series.
2. There exist series that are not equal to the sum of its Taylor Series.
Example 1. Find the Maclaurin Series of the function f (x) = ex and its radius of
convergence.
Example 2. Find the Maclaurin series for g(x) = sin x, and the interval of convergence
for the series.
Math 152 – Spring 2016
Section 10.7
3 of 8
Example 3. Find the Maclaurin series for h(x) = cos x, and the interval of convergence
for the series.
Common Maclaurin Series
∞
P
1
xn = 1 + x + x2 + x3 + x4 + · · ·
1−x =
Interval of Convergence
(−1, 1)
n=0
ex =
∞
P
n=0
sin x =
xn
n!
∞
P
=1+
x
1!
+
x2
2!
+
x3
3!
2n+1
∞
P
x3
3!
+
x5
5!
−
x2
2!
+
x4
4!
−
x6
6!
2n
x
=1−
(−1)n (2n)!
n=0
tan−1 x =
(−∞, ∞)
x
(−1)n (2n+1)!
=x−
n=0
cos x =
+ ···
∞
P
2n+1
x
=x−
(−1)n (2n+1)
n=0
x3
3
+
x5
5
x7
7!
+ ···
(−∞, ∞)
+ ···
(−∞, ∞)
x7
7
(−1, 1)
−
+ ···
Example 4. Find the Maclaurin series for the function f (x) = x2 ex , and the interval
of convergence for the series.
Math 152 – Spring 2016
Section 10.7
4 of 8
Example 5. Find the Taylor series for f (x) = ex centered at x = −5.
Example 6. Represent g(x) = sin x as the sum of its Taylor series centered at π/6.
Math 152 – Spring 2016
Example 7. (a) Evaluate
(b) Evaluate
R1
e−x
2
/2
Section 10.7
R
e−x
2
/2
dx as an infinite series.
dx to within an error of 0.001.
0
Example 8. Evaluate lim
x→0
x−3 arctan x
.
x4
5 of 8
Math 152 – Spring 2016
Section 10.7
6 of 8
Note: Power series can be added, subtracted, multiplied and divided like regular
polynomials.
Example 9. Find the first three nonzero terms in the Maclaurin series for the following.
(a) g(x) = ex cos x
(b) f (x) = tan x
Math 152 – Spring 2016
Section 10.7
7 of 8
Example 10. Evaluate the following power series. (Hint: The power series can be
written in the form of the Maclaurin series for one of our common functions.)
(a)
∞
P
n=0
(b)
∞
P
n=0
(c) 4 −
(−1)3n+1 22n+1 x6n+3
(2n+1)!
(ln 8)n
3n n!
4π 2
2!
+
16π 4
4!
−
64π 6
6!
+
256π 8
8!
− ···
Math 152 – Spring 2016
Section 10.7
8 of 8
Note. The theorem says that IF f (x) has a power series centered at a, then that power
series is the Taylor series:
f (x) =
∞
X
f (n) (a)
(x − a)n
n!
n=0
We need to determine when a function DOES actually equal its Taylor series, i.e. when
does
∞
X
f (n) (a)
f (x) =
(x − a)n
n!
n=0
Definition. Define the nth-degree Taylor polynomial Tn by
Tn (x) =
n
X
f (i) (a)
i=0
i!
(x − a)i
Also, define Rn to be the difference in f (x) and Tn :
Rn (x) = f (x) − Tn (x).
Theorem. If f (x) = Tn (x) + Rn (x), where Tn is the nth-degree Taylor polynomial of
f at a and
lim Rn (x) = 0
n→∞
for |x−a| < R, then f is equal to the sum of its Taylor series on the interval |x−a| < R.
Taylor’s Inequality. If |f (n+1) (x)| ≤ M for |x − a| < R, then the remainder Rn (x)
of the Taylor Series satisfies the inequality
|Rn (x)| ≤
M
|x − a|n+1
(n + 1)!
for |x − a| < R
Note. The following limit will be useful:
xn
=0
n→∞ n!
lim
for every real number x
Example 11. Prove that sin x is equal to the sum of its Maclaurin Series.