Week 2 Notes(1)(1)

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Representation of Functions as Power Series
Taylor and Maclaurin Series
Definition
Let f be a function such that for all n 2 N, f (n) exists on some
open interval containing a. The Taylor series of f centered at a is
the series
1
X
f (n) (a)
(x a)n .
n!
In the previous examples, we saw that some elementary functions
can be represented using a power series (at least on some open
interval).
series representation
PTo find a power
f (x) =
cn (x a)n for a function f in general, we first assume
that:
I
I
n=0
The Maclaurin series of f is the Taylor series of f centered at 0:
f is infinitely di↵erentiable, that is, f has derivatives of all
orders, on an open interval containing a, and;
1
X
f (n) (0)
n=0
f has such a power series representation.
n!
x n.
Remark
It can be shown that if f has a power series representation on
some open interval containing a, then this power series must be
the Taylor series.
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Taylor Series Does Not Always Represent a Function
Example
(a) The Maclaurin series of f (x) :=
MAT1002 Week 2
ex
is
1
X
xn
n!
n=0
1
X
(b) The Maclaurin series of f (x) := cos x is
n=0
If a power series representation exists for an infinitely di↵erentiable
function, then it must be a Taylor series. But some function may
not have a power series representation in the first place.
.
( 1)n x 2n
.
(2n)!
Consider the following function and its graph.
The following figure shows some partial sums of the Maclaurin
series of f (x) := cos x.
This function satisfies f (n) (0) = 0 for all n, so its Taylor series
centered at 0 is just the zero function. Therefore f does not equal
its Taylor series centered at 0 on any open interval containing 0.
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Taylor’s Theorem
How do we know that a function can be represented by its Taylor
series?
Theorem (Taylor’s Theorem)
Definition
Suppose that f is a function such that f , f 0 , . . ., f (n) are
continuous on [a, b], and f (n) is di↵erentiable on (a, b). Then there
exists c 2 (a, b) such that
!
n
X
f (k) (a)
f (n+1) (c)
k
f (b) =
(b a)
+
(b a)n+1
k!
(n + 1)!
f (n)
Let f be a function such that for all n 2 {0, 1, . . . , N},
exists
on some open interval containing a. The Taylor polynomial of f
(of order n) centered at a is the polynomial
Pn (x) :=
n
X
f (k) (a)
k=0
k!
(x
a)k .
k=0
= Pn (b) +
Remark
f (n+1) (c)
(b
(n + 1)!
a)n+1 .
A Taylor polynomial is a partial sum of a Taylor series.
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f (n+1) (c)
(x
(n + 1)!
|Rn (x)|  M
f (x) =
Note that if Rn (x) ! 0 as n ! 1 for all x 2 I , then
n!1
n!1
n=0
1
X
f (n) (a)
n=0
1
X
f (n) (a)
n!
(x
|x a|n+1
,
(n + 1)!
so Rn (x) ! 0 as n ! 1. In this case,
a)n+1 for some c between a and x.
f (x) = lim (Pn (x) + Rn (x)) = lim Pn (x) =
18
In particular, for a given x, if there is a positive constant M such
that |f (n+1) (t)|  M for all t between a and x, then
Suppose that f is infinitely di↵erentiable on an open interval I
containing a, and let Rn (x) := f (x) Pn (x). Then Taylor’s
theorem states that
Rn (x) =
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n!
(x
a)n .
Example
a)n .
(a) Show that the Maclaurin series of f (x) := e x converges to
f (x) for all x 2 R.
(b) Show that the Maclaurin series of f (x) := sin x converges to
f (x) for all x 2 R.
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Binomial Series
Definition
For any m 2 R and n 2 N, define the binomial coefficient m
n as
follows:
✓ ◆
✓ ◆
m
m
m(m 1)(m 2) . . . (m n + 1)
:= 1,
:=
if n 1.
0
n
n!
Remark
It can be shown that for x 2 ( 1, 1),
(1 + x)m =
1 ✓ ◆
X
m
n
n=0
x n.
Such a series is called a binomial series.
Example
Find the Maclaurin series of f (x) :=
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Limits of Indeterminate Forms
Taylor series can sometimes be used to find limits of indeterminate
forms.
Example
Evaluate
lim
x!1
lnx
x
1
using Taylor series.
More examples can be found in Chapter 10.10 in the book.
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MAT1002 Week 2
p
1 + x.
22
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