c Dr Oksana Shatalov, Fall 2012
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10.7: Taylor and Maclaurin Series
Problem: Assume that a function f (x) has a power series representation about x = a:
f (x) =
∞
X
cn (x − a)n
n=0
and f (x) has derivatives of every order. Find formula for cn in terms of f .
Solution. We have
f (x) =
Putting x = a in f (x), we get
f 0 (x) =
Substituting x = a we have
Similarly,
f 00 (x) =
then x = a ⇒ f 00 (a) =
and
f 000 (x) =
then x = a ⇒ f 00 (a) =
and
Continuing in this manner, you can see the pattern:
f (n) (a) =
The Taylor series for f (x) about x = a:
∞
X
f (n) (a)
f (x) =
(x − a)n =
n!
n=0
f 00 (a)
f 000 (a)
= f (a) + f 0 (a)(x − a) +
(x − a)2 +
(x − a)3 + . . .
2!
3!
c Dr Oksana Shatalov, Fall 2012
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Split the Taylor series as follows:
∞
X
f (n) (a)
n=0
n!
N
X
f (n) (a)
(x − a)n =
n=0
n!
(x − a)n
{z
}
TN (x)
N − th degree
Taylor polynomial
|
+
∞
X
f (n) (a)
(x − a)n
n!
n=N +1
|
{z
}
RN (x)
Remainder
THEOREM 1. If lim RN (x) = 0 when |x − a| < R then
n→∞
f (x) =
∞
X
f (n) (a)
n=0
n!
(x − a)n ,
|x − a| < R.
REMARK 2. In all examples that we will be looking at, we assume that f (x) has a power series
expansion, i.e.
lim RN (x) = 0 for some R. (This means you don’t need to show it.)
n→∞
EXAMPLE 3. Given that function f has power series expansion (i.e. Taylor series) centered at 4. Find
this expansion and its radius of convergence if it is given that
f
(n)
(4) =
(−1)n−1 1 · 3 · 5 · . . . · (2n − 3)
23n−1 n
c Dr Oksana Shatalov, Fall 2012
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EXAMPLE 4. Find Taylor series for f (x) = e3x at x = 1. What is the associated radius of convergence?
EXAMPLE 5. Find Taylor series for f (x) = ln x at x = 1. What is the associated radius of convergence?
c Dr Oksana Shatalov, Fall 2012
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EXAMPLE 6. Find Taylor series for ln(1 + x) centered at x = 0. What is the associated radius of
convergence?
The Maclaurin series is the Taylor series about x = 0 (i.e. a=0):
f (x) =
∞
X
f (n) (0)
n=0
n!
xn = f (0) + f 0 (0)x +
EXAMPLE 7. Find the Maclaurin series for f (x):
(a) f (x) = ex
(b) f (x) = e−x
(c) f (x) = x5 e−2x
2
f 00 (0) 2 f 000 (0) 3
x +
x + ...
2!
3!
c Dr Oksana Shatalov, Fall 2012
EXAMPLE 8. Find the Maclaurin series for f (x):
(a) f (x) = cos x
(b) f (x) = sin x
5
c Dr Oksana Shatalov, Fall 2012
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Known Mclaurin series and their intervals of convergence you must have memorized:
1
∞
X
=
1−x
xn
=
1 + x + x2 + x3 + . . .
(1, 1)
=
1+x+
x2 x3
+
+ ...
2!
3!
(−∞, ∞)
n=0
ex
∞
X
xn
=
n=0
cos x
∞
X
(−1)n x2n
=
n=0
sin x
=
(2n)!
∞
X
(−1)n x2n+1
n=0
arctan x =
n!
∞
X
(2n + 1)!
(−1)n
n=0
x2n+1
2n + 1
= 1−
x2 x4 x6
+
−
+ . . . (−∞, ∞)
2
4!
6!
= x−
x3 x5 x7
+
−
+ . . . (−∞, ∞)
3!
5!
7!
= x−
EXAMPLE 9. Find the sum of the series:
(a)
∞
X
(−1)n π 2n
n=0
(b)
∞
X
2012n
n=0
(c)
32n (2n!)
∞
X
n!
(−1)n
n=0
x4n+2
2n + 1
x3
3
+
x5
5
−
x7
7
+ ···
[−1, 1]
c Dr Oksana Shatalov, Fall 2012
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Z
EXAMPLE 10. (a) Determine Maclaurin Series for
Z
(b) Evaluate
0
1
sin x
dx
x
sin x
dx correct to within an error of 0.001.
x
c Dr Oksana Shatalov, Fall 2012
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EXAMPLE 11. Use series to find the limit: lim
x→0
1 − cos x
1 + x − ex