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Math 152-copyright Joe Kahlig, 13A
Section 10.7: Taylor and Maclaurin Series
Definition: If a function has a power series representation, then this power series is referred to as the
Taylor series of the function f at a (or about a or centered at a). If this series is centered at x = 0,
then this series is given the special name Maclaurin series.
Theorem: If f (x) has a power series representation at a, i.e.
f (x) =
∞
X
n=0
centered at x = a, that is if
cn (x − a)n with |x − a| < R then its coefficients are given by the formula
cn =
f (x) = c0 + c1 (x − a) + c2 (x − a)2 + c3 (x − a)3 + c4 (x − a)4 + ...
Example: Find the Maclaurin series and the radius of convergence for f (x) = e2x
Math 152-copyright Joe Kahlig, 13A
Example: Find the Maclaurin series and the radius of convergence for f (x) = sin(2x)
Example: Find the Maclaurin series and the radius of convergence for f (x) = cos(2x)
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Math 152-copyright Joe Kahlig, 13A
Important Maclaurin series
∞
X
1
=
xn = 1 + x + x2 + x3 + ...
1−x
n=0
ex =
∞
X
xn
n=0
sin(x) =
n!
=1+x+
∞
X
(−1)n x2n+1
n=0
cos(x) =
x2
+ ...
2!
(2n + 1)!
∞
X
(−1)n x2n
n=0
tan−1 (x) =
=1−
∞
X
(−1)n x2n+1
n=0
ln(1 + x) =
(2n)!
=x−
2n + 1
∞
X
(−1)n xn+1
n=0
n+1
|x| < 1
R=∞
x3 x5
+
− ...
3!
5!
x2 x4
+
− ...
2!
4!
=x−
=x−
R=∞
R=∞
x3 x5
+
− ...
3
5
x2 x3
+
− ...
2
3
|x| ≤ 1
−1 < x ≤ 1
Example: Find the Maclaurin series and the radius of convergence for f (x) = tan−1 (2x3 )
1+x
Example: Find the Maclaurin series and the radius of convergence for f (x) = ln
1−x
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Math 152-copyright Joe Kahlig, 13A
Example: Find the Taylor series of f (x) = sin(x) at x =
π
6
Example: Find the Taylor series of f (x) = ln(x) about a = 2
Math 152-copyright Joe Kahlig, 13A
1
Example: Find the Taylor series of f (x) = √ about a = 4
x
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Math 152-copyright Joe Kahlig, 13A
Example: Use series to evaluate this limit.
lim
x→0
1 − cos(x)
1 + x − ex
Example: Find the sum of this series.
∞
X
(−1)n π 2n
n=0
32n (2n)!
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Math 152-copyright Joe Kahlig, 13A
Example: Find the first three non-zero terms in the Maclaurin series for f (x) = ex sin(x)
ex sin(x) =
!
x2 x3
+
+ .....
1+x+
2!
3!
!
x3 x5
x−
+
− ...
3!
5!
Example: Find the first three non-zero terms in the Maclaurin series for f (x) = tan(x) =
sin(x)
cos(x)
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Math 152-copyright Joe Kahlig, 13A
Taylor Polynomials.
The Taylor series of a function, f (x), can be expressed: f (x) =
∞
X
f (k)(a)
k=0
gree Taylor polynomial of f (x) at a, denoted Tn is given by
Tn (x) =
n
X
f (k) (a)
k=0
k!
(x − a)k = f (a) + f ′ (a)(x − a) +
k!
(x − a)k . The n-th de-
f ′′ (a)
f (n) (a)
(x − a)2 + ... +
(x − a)n
2!
n!
Define Rn (x) to be the remainder of the series such that f (x) = Tn (x) + Rn (x)
Rn (x) =
∞
X
k=n+1
f (k) (a)
(x − a)k
k!
The following graph shows the function
x2
1
and T0 , T2 , T4 , and T6 .
+ 25
Theorem: If f (x) = Tn (x) + Rn (x), where Tn is the nth-degree Taylor polynomial of f at a and
lim Rn = 0 for |x−a| < R, then f (x) is equal to the sum of its Taylor series on the interval |x−a| < R.
n→∞
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