<
Section
3.1
T
a
ylo
r
P
olynomials
Goals:
1
App
roximate
a
function
with
a
T
a
ylor
p
olynomial.
2
Compute
error
b
ounds
fo
r
a
T
a
ylor
p
olynomial.
Question
3.1.1
Ho
w
Can
W
e
Imp
rove
on
a
Linea
rization?
F
o
rmula
The
linearization
or
tangent
line
to
a
function
f
(
x
)
at
a
has
the
equation.
L
(
x
)
=
f
(
a
)
+
f
′
(
a
)(
x
−
a
)
By
design
f
and
L
have
1
Equal
values
at
a
.
2
Equal
first
derivatives
at
a
.
W
e
could
make
a
b
etter
app
roximation,
if
we
could
match
second,
third,
fourth
derivatives
of
f
(
x
).
A
line
cannot
do
that,
but
a
p
olynomial
c
an.
165
Question
3.1.2
What
Is
a
T
a
ylo
r
P
olynomial?
Definition
The
n
th
T
aylo
r
p
olynomial
of
f
(
x
)
at
x
=
a
is
a
degree
n
p
olynomial
that
shares
the
value
and
first
n
derivatives
of
f
at
x
=
a
.
Its
formula
is
T
n
(
x
)
=
n
X
k
=0
f
(
k
)
(
a
)
k
!
(
x
−
a
)
k
.
Rema
rks
The
variable
is
x
.
f
(
k
)
(
a
)
is
not
a
function
but
a
number.
f
(0)
is
the
zeroth
derivative,
meaning
f
(0)
(
a
)
=
f
(
a
).
0!
is
defined
to
b
e
1.
166
Example
3.1.3
Computing
a
T
a
ylo
r
P
olynomial
a
Find
the
degree
3
T
a
ylor
p
olynomial
of
y
=
√
x
at
x
=
4.
b
Use
it
to
estimate
√
5.
167
Example
3.1.4
W
riting
a
Sum
in
Σ
Notation
W
rite
each
of
the
following
sums
in
Σ
notation.
a
4
+
7
+
10
+
13
+
16
+
19
+
22
b
2
+
6
+
18
+
54
+
162
+
486
c
−
3
+
4
−
5
+
6
−
7
+
8
−
9
+
10
d
1
4
+
√
2
9
+
√
3
16
+
2
25
+
√
5
36
168
Example
3.1.5
A
T
a
ylo
r
P
olynomial
in
P
Notation
W
rite
the
10th
degree
T
aylo
r
P
olynomial
for
f
(
x
)
=
1
x
2
centered
at
x
=
3.
169
Question
3.1.6
Ho
w
Accurate
Is
the
T
a
ylo
r
P
olynomial?
Theo
rem
Let
f
and
g
b
e
differentiable
functions.
Consider
an
interval
[
a
,
b
],
and
supp
ose
f
(
a
)
=
g
(
a
).
1
If
f
′
(
x
)
=
g
′
(
x
)
on
[
a
,
b
],
then
f
(
x
)
=
g
(
x
)
on
[
a
,
b
]
2
If
f
′
(
x
)
<
g
′
(
x
)
on
[
a
,
b
],
then
f
(
x
)
<
g
(
x
)
on
(
a
,
b
]
Reasoning
Intuitive
If
tw
o
functions
sta
rt
at
the
same
value
at
a
,
then
the
one
that
grows
faster
will
have
a
higher
value
at
b
.
Fo
rmal
The
Fundamental
Theo
rem
of
Calculus
sa
ys
f
(
x
)
−
f
(
a
)
=
Z
x
a
f
′
(
t
)
dt
g
(
x
)
−
g
(
a
)
=
Z
x
a
g
′
(
t
)
dt
.
La
rger
functions
have
la
rger
integrals.
170
Question
3.1.6
How
Accurate
Is
the
T
aylo
r
Polynomial?
Figure:
Tw
o
functions
with
a
common
value
at
a
:
f
(
x
)
with
a
smaller
derivative
and
g
(
x
)
with
a
la
rger
derivative.
171
Question
3.1.6
How
Accurate
Is
the
T
aylo
r
Polynomial?
Notation
Given
a
function
f
(
x
)
and
its
n
th
T
aylo
r
p
olynomial
T
n
(
x
)
centered
at
a
,
the
remainder
at
x
is
R
n
(
x
)
=
f
(
x
)
−
T
n
(
x
)
If
we
are
using
T
n
(
x
)
to
app
roximate
f
(
x
),
R
n
(
x
)
=
−
erro
r
of
T
n
(
x
)
.
W
e
should
b
e
very
interested
in
knowing
the
value
of
R
n
(
x
).
We
will
use
our
derivative
compa
rison
theo
rem
to
mak
e
t
w
o
arguments
1
If
f
(
n
+1)
(
x
)
is
a
constant
M
,
then
we
can
compute
R
n
(
x
)
exactly
.
2
If
|
f
(
n
+1)
(
x
)
|
≤
M
then
the
erro
r
in
1
is
the
w
orst-case
scena
rio.
172
Question
3.1.6
How
Accurate
Is
the
T
aylo
r
Polynomial?
Theo
rem
If
f
(
n
+1)
(
x
)
is
a
constant
M
on
[
a
,
b
],
then
f
(
x
)
=
T
n
+1
(
x
)
=
T
n
(
x
)
+
M
(
n
+
1)!
(
x
−
a
)
n
+1
.
d
dx
n
+1
f
(
x
)
=
d
dx
n
+1
T
n
+1
(
x
)
=
M
on
[
a
,
b
]
d
dx
n
f
(
a
)
=
d
dx
n
T
n
+1
(
a
)
d
dx
n
f
(
x
)
=
d
dx
n
T
n
+1
(
x
)
on
[
a
,
b
]
d
dx
n
−
1
f
(
a
)
=
d
dx
n
−
1
T
n
+1
(
a
)
d
dx
n
−
1
f
(
x
)
=
d
dx
n
−
1
T
n
+1
(
x
)
on
[
a
,
b
]
d
dx
f
(
a
)
=
d
dx
T
n
+1
(
a
)
d
dx
f
(
x
)
=
d
dx
T
n
+1
(
x
)
on
[
a
,
b
]
f
(
a
)
=
T
n
+1
(
a
)
f
(
x
)
=
T
n
+1
(
x
)
on
[
a
,
b
]
Because
derivatives
and
values
of
a
T
aylor
polynomial
match
the
function
173
Question
3.1.6
How
Accurate
Is
the
T
aylo
r
Polynomial?
But
what
if
f
(
n
+1)
(
x
)
is
not
a
constant?
In
this
case
w
e
will
settle
fo
r
a
b
ound
on
f
(
n
+1)
(
x
).
Theo
rem
(T
a
ylor’s
Inequalit
y)
If
f
(
n
+1)
(
t
)
≤
M
fo
r
all
x
betw
een
a
and
b
,
then
for
all
x
betw
een
a
and
b
,
|
R
n
(
x
)
|
≤
M
(
n
+
1)!
(
x
−
a
)
n
+1
174
Question
3.1.6
How
Accurate
Is
the
T
aylo
r
Polynomial?
T
o
prove
T
aylo
r’s
Inequality
,
w
e
compare
the
derivatives
of
f
(
x
)
with
the
w
orst-case
scena
rio
w
(
x
)
=
T
n
(
x
)
+
M
(
n
+1)!
(
x
−
a
)
n
+1
.
d
dx
n
+1
f
(
x
)
≤
d
dx
n
+1
w
(
a
)
=
M
on
[
a
,
b
]
d
dx
n
f
(
a
)
=
d
dx
n
w
(
a
)
d
dx
n
f
(
x
)
≤
d
dx
n
w
(
x
)
on
[
a
,
b
]
d
dx
n
−
1
f
(
a
)
=
d
dx
n
−
1
w
(
a
)
d
dx
n
−
1
f
(
x
)
≤
d
dx
n
−
1
w
(
x
)
on
[
a
,
b
]
d
dx
f
(
a
)
=
d
dx
w
(
a
)
d
dx
f
(
x
)
≤
d
dx
w
(
x
)
on
[
a
,
b
]
f
(
a
)
=
w
(
a
)
f
(
x
)
≤
w
(
x
)
on
[
a
,
b
]
Because
M
is
a
b
ound
on
d
dx
n
+1
f
(
x
)
Because
derivatives
and
values
of
a
T
aylor
polynomial
match
the
function
175
Question
3.1.6
How
Accurate
Is
the
T
aylo
r
Polynomial?
T
o
finish
the
argument
we
need
to
1
Pro
duce
a
low
er
b
ound
fo
r
f
using
w
(
x
)
=
T
n
(
x
)
−
M
(
n
+1)!
(
x
−
a
)
n
+1
.
2
Solve
the
inequalit
y
bounds
fo
r
R
n
(
x
).
T
n
(
x
)
−
M
(
n
+
1)!
(
x
−
a
)
n
+1
≤
f
(
x
)
≤
T
n
(
x
)
+
M
(
n
+
1)!
(
x
−
a
)
n
+1
−
M
(
n
+
1)!
(
x
−
a
)
n
+1
≤
R
n
(
x
)
≤
M
(
n
+
1)!
(
x
−
a
)
n
+1
3
Rep
eat
for
intervals
of
the
fo
rm
[
b
,
a
].
These
w
ork
the
same
w
ay
with
a
sign
reversed.
176
Example
3.1.7
A
T
a
ylo
r
App
ro
ximation
Erro
r
Bound
Let
f
(
x
)
=
sin
x
.
a
Give
a
general
form
for
the
n
th
T
a
ylor
p
olynomial
for
f
at
x
=
0.
b
Find
a
b
ound
on
f
(
n
)
(
x
)
for
each
n
.
c
What
happ
ens
to
the
erro
r
bound
as
x
increases
but
n
stays
the
same?
d
What
happ
ens
to
the
erro
r
bound
as
n
increases
but
x
stays
the
same?
e
What
do
es
this
tell
us
ab
out
the
relationship
b
et
ween
the
T
n
(
x
)
app
roximations
and
f
(
x
)?
177
Example
3.1.7
A
T
a
ylo
r
App
ro
ximation
Erro
r
Bound
Let
f
(
x
)
=
sin
x
.
a
Give
a
general
form
for
the
n
th
T
a
ylor
p
olynomial
for
f
at
x
=
0.
177
Example
3.1.7
A
T
a
ylo
r
App
ro
ximation
Erro
r
Bound
Let
f
(
x
)
=
sin
x
.
b
Find
a
b
ound
on
f
(
n
)
(
x
)
for
each
n
.
177
Example
3.1.7
A
T
a
ylo
r
App
ro
ximation
Erro
r
Bound
Let
f
(
x
)
=
sin
x
.
c
What
happ
ens
to
the
erro
r
bound
as
x
increases
but
n
stays
the
same?
177
Example
3.1.7
A
T
a
ylo
r
App
ro
ximation
Erro
r
Bound
Let
f
(
x
)
=
sin
x
.
d
What
happ
ens
to
the
erro
r
bound
as
n
increases
but
x
stays
the
same?
177
Example
3.1.7
A
T
a
ylo
r
App
ro
ximation
Erro
r
Bound
Let
f
(
x
)
=
sin
x
.
e
What
do
es
this
tell
us
ab
out
the
relationship
b
et
ween
the
T
n
(
x
)
app
roximations
and
f
(
x
)?
177
Example
3.1.7
A
T
a
ylor
Approximation
Error
Bound
Click to Load Applet
Figure:
f
(
x
)
=
sin
x
app
roximated
by
its
T
aylo
r
p
olynomials,
T
n
(
x
)
178
Example
3.1.7
A
T
a
ylor
Approximation
Error
Bound
Main
Ideas
In
order
to
understand
how
the
erro
r
changes
as
n
increases,
w
e
need
to
have
an
exp
ression
fo
r
f
(
n
)
(
x
).
W
e
can
cho
ose
M
to
b
e
the
la
rgest
value
of
|
f
(
n
+1)
|
on
the
interval
[
a
,
x
].
This
may
not
be
the
value
of
|
f
(
n
+1)
(
a
)
|
.
In
general,
T
a
ylor
p
olynomials
will
b
ecome
less
accurate
the
fa
rther
y
ou
get
from
a
.
W
e
can
often
mitigate
this
inaccuracy
by
cho
osing
larger
values
of
n
.
179
Example
3.1.7
A
T
a
ylor
Approximation
Error
Bound
Some
functions
a
re
not
well
estimated
by
their
T
aylo
r
p
olynomial.
Example
f
(
x
)
=
(
0
if
x
≤
0
e
−
1
x
if
x
>
0
f
(
k
)
(0)
=
0
for
all
k
.
So
the
T
a
ylor
p
olynomial
at
x
=
0
is
T
n
(
x
)
=
n
X
k
=0
0
x
k
.
No
matter
how
la
rge
n
gets,
T
n
(
x
)
will
not
get
any
closer
to
f
(
x
)
fo
r
any
x
>
0.
180
Example
3.1.7
A
T
a
ylor
Approximation
Error
Bound
Ho
w
can
this
happen,
given
T
aylo
r’s
Inequality?
The
derivatives
of
f
get
bigger
and
bigger.
M
gro
ws
so
fast
that
the
erro
r
R
n
(
x
)
gets
no
smaller.
Click to Load Applet
Figure:
A
function
whose
derivative
bounds
grow
factorially
181
Section
3.1
Summa
ry
Questions
Q1
Why
do
w
e
use
T
aylo
r
p
olynomials?
Q2
Why
is
there
a
denominato
r
of
k
!
in
the
formula
for
a
T
aylo
r
p
olynomial?
Q3
Explain
why
w
e’d
alw
a
ys
rather
center
a
T
a
ylor
p
olynomial
fo
r
y
=
ln
x
at
x
=
1.
Q4
What
properties
mak
e
a
T
aylo
r
p
olynomial
T
n
(
x
)
a
b
etter
app
roximation
of
f
(
x
)?
182
Section
3.1
Q6
Supp
ose
you
were
lo
cked
in
a
ro
om
with
only
a
p
encil
and
pap
er
and
ask
ed
to
compute
the
first
ten
decimal
places
of
the
following
numb
ers:
4
17
√
7
e
Which
could
y
ou
compute?
F
or
the
ones
y
ou
can
compute,
ho
w
would
you
do
it?
183
Section
3.1
Q28
Let
f
(
x
)
=
xe
x
.
a
Compute
the
T
a
ylor
p
olynomial
T
3
(
x
)
for
f
(
x
)
centered
at
x
=
0.
b
Compute
the
theo
retical
erro
r
b
ound
fo
r
T
3
(2).
c
Explain
the
difficulties
that
would
a
rise
from
this
error
b
ound,
if
y
our
goal
is
to
appro
ximate
f
(2)
b
y
hand.
Can
you
resolve
them?
184
Section
3.1
Q30
Consider
the
graph
of
y
=
f
(
x
)
b
elow.
a
Supp
ose
you
wanted
to
p
roduce
the
second
degree
T
aylo
r
p
olynomial
of
f
centered
at
a
=
−
1.
Indicate
whether
the
constant
term
and
each
co
efficient
w
ould
be
p
ositive
or
negative.
Provide
evidence
for
your
answ
er.
b
W
ould
T
2
(4)
underestimate
o
r
overestimate
f
(4)?
Explain.
185
Section
3.2
Sequences
Goals:
1
Use
notation
to
describ
e
the
terms
of
an
infinite
sequence.
2
Calculate
the
limit
of
an
infinite
sequence.
Question
3.2.1
What
Is
a
Sequence?
A
sequence
is
an
o
rdered
set
of
numbers.
If
this
set
is
infinite,
we
can
most
rigorously
define
it
by
giving
a
general
fo
rmula
for
the
n
th
term
for
some
index
va
riable
n
.
He
re
are
three
different
notations
fo
r
the
same
sequence.
1
2
,
2
3
,
3
4
,
4
5
.
.
.
n
n
+
1
∞
n
=1
a
n
=
n
n
+
1
187
Question
3.2.2
What
Is
the
Limit
of
a
Sequence?
Definition
If
we
can
mak
e
the
elements
of
a
sequence
a
n
a
rbitrarily
close
to
some
numb
er
L
b
y
considering
only
n
above
a
certain
numb
er,
then
w
e
write
lim
n
→∞
a
n
=
L
and
we
say
the
sequence
converges
to
L
.
If
a
n
do
es
not
converge
to
any
such
L
then
we
say
it
diverges
.
188
Question
3.2.2
What
Is
the
Limit
of
a
Sequence?
Rema
rks
The
first
few
or
even
the
first
thousand
terms
of
a
sequence
have
no
b
ea
ring
on
the
limit.
We
only
care
that
w
e
can
eventually
get
close
to
L
.
“Arbitra
rily
close”
means
any
level
of
closeness
than
anyone
could
ask
for.
Eventually
the
sequence
must
b
e
within
1
100
of
L
,
and
1
1000
and
1
1000000
.
189
Question
3.2.2
What
Is
the
Limit
of
a
Sequence?
Click to Load Applet
Figure:
A
sequence
converging
to
L
=
3
190
Example
3.2.3
Computing
a
Limit
Calculate
lim
n
→∞
n
n
+
1
191
Example
3.2.3
Computing
a
Limit
Calculate
lim
n
→∞
n
n
+
1
1
2
,
2
3
,
3
4
,
4
5
.
.
.
191
Example
3.2.3
Computing
a
Limit
Click to Load Applet
Figure:
The
sequence
n
n
+1
converges
to
L
=
1.
192
Question
3.2.4
Ho
w
Are
Limits
of
Sequences
and
F
unctions
Related?
Theo
rem
Supp
ose
for
a
sequence
a
n
,
there
is
a
function
f
(
x
)
such
that
f
(
n
)
=
a
n
fo
r
all
n
(or
at
least
all
n
sufficiently
la
rge).
If
lim
x
→∞
f
(
x
)
=
L
w
e
can
conclude
that
lim
n
→∞
a
n
=
L
.
193
Example
3.2.5
Sequence
Limits
Using
F
unctions
Find
limits
of
the
follo
wing
sequences:
a
lim
n
→∞
2
n
n
+
3
b
lim
n
→∞
1
n
3
c
lim
n
→∞
e
−
n
d
lim
n
→∞
n
2
e
n
e
lim
n
→∞
(
−
1)
n
194
Example
3.2.5
Sequence
Limits
Using
Functions
Theo
rems
All
of
the
laws
for
limits
of
functions
at
infinity
also
apply
to
limits
of
sequences.
Fo
r
instance
supp
ose
lim
n
→∞
a
n
=
L
and
lim
n
→∞
b
n
=
M
.
If
c
n
=
a
n
+
b
n
then
lim
n
→∞
c
n
=
L
+
M
.
195
Synthesis
3.2.6
Indeterminate
F
o
rms
with
F
acto
rials
Dominance
W
e
sa
y
f
(
x
)
dominates
g
(
x
)
if
lim
x
→∞
f
(
x
)
g
(
x
)
=
±∞
.
We
write
f
(
x
)
>>
g
(
x
)
Even
if
y
ou
include
a
constant
multiple
o
r
add
multiple
functions
together,
the
dominant
function
will
outgrow
any
combination
of
dominated
ones.
W
e
have
already
established
an
o
rder
of
dominance
using
l’Hˆ
opital’s
rule:
exp
onential
(larger
base
>>
smaller
base)
>>
p
olynomial
(larger
degree
>>
smaller)
>>
ro
ot
(smaller
pow
er
>>
la
rger)
>>
loga
rithm
(smaller
base
>>
la
r
ger)
196
Synthesis
3.2.6
Indeterminate
Fo
rms
with
Facto
rials
Theo
rem
As
n
→
∞
,
n
!
will
eventually
dominate
any
exponential
function
(and
thus
any
p
olynomial,
ro
ot
o
r
loga
rithm).
197
Section
3.2
Summa
ry
Questions
Q1
Why
do
w
e
use
n
instead
of
x
as
an
index
fo
r
a
sequence?
Q2
Describ
e
three
different
wa
ys
of
denoting
a
sequence.
Q3
When
is
the
limit
of
a
sequence
equal
to
the
limit
of
a
function?
Q4
If
a
n
=
b
n
+
1000
fo
r
1
≤
n
≤
2000000,
what
do
es
that
tell
us
ab
out
the
limits
lim
n
→∞
a
n
and
lim
n
→∞
b
n
?
198
Section
3.2
Q12
Consider
the
sequence
a
n
=
n
sin(
π
n
)
a
What
is
lim
x
→∞
x
sin(
π
x
)?
b
Compute
the
first
few
values
a
1
,
a
2
,
a
3
,
and
a
4
.
c
What
is
lim
n
→∞
n
sin(
π
n
)?
d
Do
es
this
contradict
one
of
our
theo
rems?
Explain.
199
Section
3.2
Q12
Rema
rk
Read
theorems
carefully
.
The
hyp
othesis
and
conclusion
a
re
not
interchangeable.
Mixing
them
up
can
turn
a
true
theo
rem
into
a
false
one.
199
Section
3.2
Q24
Let
T
n
(
x
)
b
e
the
n
th
T
aylo
r
p
olynomial
of
f
(
x
)
=
ln
x
centered
at
x
=
1.
a
W
rite
an
expression
fo
r
T
n
(
x
)
using
Σ
notation.
b
W
rite
an
expression
fo
r
the
error
b
ound
of
T
n
(
x
)
for
some
x
b
et
ween
0
and
1.
c
F
or
what
values
of
x
will
the
error
b
ound
shrink
to
0
as
n
goes
to
∞
?
200
Section
3.2
Q24
a
W
rite
an
expression
fo
r
T
n
(
x
)
using
Σ
notation.
200
Section
3.2
Q24
b
W
rite
an
expression
fo
r
the
error
b
ound
of
T
n
(
x
)
for
some
x
b
et
ween
0
and
1.
200
Section
3.2
Q24
c
F
or
what
values
of
x
will
the
error
b
ound
shrink
to
0
as
n
goes
to
∞
?
200
Section
3.3
Series
Goals:
1
Identify
partial
sums
of
a
series.
2
Recognize
harmonic
and
alternating
harmonic
series.
3
Apply
the
divergence
test.
4
Evaluate
geometric
series.
5
Apply
the
ratio
test.
Question
3.3.1
What
Is
a
Series?
Y
ou
have
b
een
encountering
series
since
y
ou
first
learned
ab
out
decimals.
Y
ou
likely
have
not
seen
a
rigorous
description
of
what
they
mean.
0
.
33333333
.
.
.
3
.
1415926
...
W
e
can
write
0
.
3333
.
.
.
=
3
10
+
3
100
+
3
1000
+
3
10000
+
·
·
·
o
r
3
.
1415
.
.
.
=
3
+
1
10
+
4
100
+
1
1000
+
5
10000
+
·
·
·
Y
ou
may
have
an
intuitive
sense
of
what
these
quantities
are,
but
what
do
es
it
mean
to
add
up
infinitely
many
numb
ers?
202
Question
3.3.1
What
Is
a
Series?
Definition
A
series
is
a
sum
of
the
form
∞
X
k
=1
a
k
where
a
k
is
an
infinite
sequence.
If
it
is
mo
re
convenient,
we
can
give
k
a
different
initial
value.
If
the
context
is
clea
r,
w
e
can
write
X
a
k
as
a
sho
rthand.
Example
0
.
33333
.
.
.
=
∞
X
k
=1
3
10
k
The
harmonic
series
is
∞
X
k
=1
1
k
This
tells
us
what
a
series
is
but
not
how
to
evaluate
it.
Ho
w
do
we
kno
w
that,
for
example
0
.
333
.
.
.
=
1
3
?
203
Question
3.3.1
What
Is
a
Series?
W
e
evaluate
a
series
by
asso
ciating
it
with
a
sequence
of
pa
rtial
sums.
Definition
The
n
th
pa
rtial
sum
of
the
series
∞
X
k
=1
a
k
is
s
n
=
a
1
+
a
2
+
a
3
+
·
·
·
+
a
n
A
series
∞
X
k
=1
a
k
converges
to
L
if
lim
n
→∞
s
n
=
L
.
A
series
that
do
es
not
converge
to
any
L
diverges
.
V
o
cabulary
Note
Do
not
confuse
a
sequence
with
a
series.
One
is
a
list
of
numbers.
The
other
is
the
sum
of
a
list
of
numb
ers.
204
Example
3.3.2
Computing
P
a
rtial
Sums
Consider
∞
X
k
=1
3
10
k
.
a
Compute
the
first
few
pa
rtial
sums
s
1
,
s
2
,
s
3
of
this
series.
b
Compute
lim
n
→∞
s
n
205
Example
3.3.2
Computing
Pa
rtial
Sums
Main
Idea
Often,
we
can
show
that
∞
X
k
=1
a
k
=
L
by
computing
L
−
s
n
and
seeing
that
it
converges
to
0.
Click to Load Applet
Figure:
The
partial
sums
s
n
converging
to
L
=
1
3
206
Example
3.3.3
The
Ha
rmonic
Series
Consider
a
pa
rtial
sum
of
the
harmonic
series
∞
X
k
=1
1
k
.
s
8
=
1
1
+
1
2
+
1
3
+
1
4
+
1
5
+
1
6
+
1
7
+
1
8
>
1
1
+
1
2
+
1
4
+
1
4
|
{z
}
1
2
+
1
8
+
1
8
+
1
8
+
1
8
|
{z
}
1
2
=
1
+
1
2
+
1
2
+
1
2
In
general
w
e
can
m
a
k
e
s
n
bigger
than
a
ny
integer
c
b
y
setting
n
=
2
m
where
1
+
1
2
m
>
c
.
This
tells
us
that
the
ha
rmonic
series
diverges.
207
Example
3.3.3
The
Ha
rmonic
Series
Consider
a
pa
rtial
sum
of
the
harmonic
series
∞
X
k
=1
1
k
.
s
8
=
1
1
+
1
2
+
1
3
+
1
4
+
1
5
+
1
6
+
1
7
+
1
8
>
1
1
+
1
2
+
1
4
+
1
4
|
{z
}
1
2
+
1
8
+
1
8
+
1
8
+
1
8
|
{z
}
1
2
=
1
+
1
2
+
1
2
+
1
2
In
general
w
e
can
m
a
k
e
s
n
bigger
than
a
ny
integer
c
b
y
setting
n
=
2
m
where
1
+
1
2
m
>
c
.
This
tells
us
that
the
ha
rmonic
series
diverges.
207
Example
3.3.3
The
Ha
rmonic
Series
Consider
a
pa
rtial
sum
of
the
harmonic
series
∞
X
k
=1
1
k
.
s
8
=
1
1
+
1
2
+
1
3
+
1
4
+
1
5
+
1
6
+
1
7
+
1
8
>
1
1
+
1
2
+
1
4
+
1
4
|
{z
}
1
2
+
1
8
+
1
8
+
1
8
+
1
8
|
{z
}
1
2
=1
+
1
2
+
1
2
+
1
2
In
general
w
e
can
m
a
k
e
s
n
bigger
than
a
ny
integer
c
b
y
setting
n
=
2
m
where
1
+
1
2
m
>
c
.
This
tells
us
that
the
ha
rmonic
series
diverges.
207
Example
3.3.3
The
Ha
rmonic
Series
Consider
a
pa
rtial
sum
of
the
harmonic
series
∞
X
k
=1
1
k
.
s
8
=
1
1
+
1
2
+
1
3
+
1
4
+
1
5
+
1
6
+
1
7
+
1
8
>
1
1
+
1
2
+
1
4
+
1
4
|
{z
}
1
2
+
1
8
+
1
8
+
1
8
+
1
8
|
{z
}
1
2
=1
+
1
2
+
1
2
+
1
2
In
general
w
e
can
make
s
n
bigger
than
any
integer
c
b
y
setting
n
=
2
m
where
1
+
1
2
m
>
c
.
This
tells
us
that
the
harmonic
series
diverges.
207
Question
3.3.4
What
Is
a
Geometric
Series?
Definition
A
geometric
series
is
a
series
of
the
fo
rm
∞
X
k
=1
a
r
k
−
1
.
a
is
the
initial
term.
r
is
the
common
ratio
b
etw
een
terms.
Example
∞
X
k
=1
1
2
k
−
1
=
1
+
1
2
+
1
4
+
1
8
+
·
·
·
∞
X
k
=1
3
10
1
10
k
−
1
=
3
10
+
3
100
+
3
1000
+
·
·
·
=
1
3
208
Question
3.3.4
What
Is
a
Geometric
Series?
Evaluate
∞
X
k
=1
a
r
k
−
1
.
209
Question
3.3.4
What
Is
a
Geometric
Series?
Figure:
The
partial
sums
of
P
a
r
k
−
1
fo
r
va
rious
r
210
Question
3.3.4
What
Is
a
Geometric
Series?
Theo
rem
Geometric
series
have
the
follo
wing
pa
rtial
sums
s
n
=
n
X
k
=1
a
r
k
−
1
=
a
(1
−
r
n
)
1
−
r
if
r
=
1
an
if
r
=
1
These
converge
to
a
1
−
r
when
|
r
|
<
1
and
diverge
when
|
r
|
≥
1.
211
Example
3.3.5
Evaluating
Geometric
Series
Identify
a
and
r
in
the
follo
wing
geometric
series.
Then
evaluate
the
series.
a
2
3
+
4
15
+
8
75
+
·
·
·
b
∞
X
n
=2
3
n
c
0
.
999999
.
.
.
212
Question
3.3.6
What
Do
es
the
Size
of
a
k
T
ell
Us
Ab
out
P
a
k
?
Theo
rem
(The
Divergence
T
est)
Let
a
k
b
e
a
sequence.
If
lim
k
→∞
a
k
=
0,
then
the
series
∞
X
k
=1
a
k
diverges.
Rema
rk
The
divergence
test
do
es
not
tell
us
anything,
if
lim
k
→∞
a
k
=
0.
The
series
might
converge,
and
it
might
not.
In
this
case
w
e
say
the
test
is
inconclusive
.
213
Example
3.3.7
Applying
the
Divergence
T
est
What
do
es
the
divergence
test
tell
us
ab
out
each
of
the
following
series?
a
∞
X
k
=2
3
k
b
∞
X
k
=2
1
k
c
∞
X
k
=2
k
2
−
1
3
k
2
+
7
d
∞
X
k
=2
k
2
e
k
214
Question
3.3.8
What
Is
the
Ratio
T
est?
The
convergence
of
a
geometric
series
dep
ends
on
the
ratio
r
.
Even
when
a
series
is
not
geometric,
w
e
can
attempt
to
apply
similar
reasoning
to
determine
whether
it
converges.
Theo
rem
(The
Ratio
T
est)
If
lim
k
→∞
a
k
+1
a
k
=
L
<
1,
then
X
a
k
converges
absolutely
.
If
lim
k
→∞
a
k
+1
a
k
=
L
>
1
or
is
infinite,
then
X
a
k
is
divergent.
If
lim
k
→∞
a
k
+1
a
k
=
1,
then
the
ratio
test
is
inconclusive.
215
Question
3.3.8
What
Is
the
Ratio
T
est?
Rema
rk
Converges
absolutely
is
a
term
for
series
with
b
oth
positive
and
negative
terms.
It
means
the
series
w
ould
converge,
even
if
the
signs
of
all
the
terms
were
all
positive.
The
alternative
is
conditional
convergence
,
meaning
the
series’s
convergence
may
require
the
p
ositive
and
negative
terms
partially
canceling
each
other
out.
Example
The
series
1
−
1
2
+
1
3
−
1
4
+
1
5
−
·
·
·
converges
(we
won’t
p
rove
this).
If
w
e
made
all
the
terms
p
ositive,
it
w
ould
be
the
harmonic
series,
which
diverges.
This
series
converges
conditionally
,
not
absolutely
.
216
Example
3.3.9
Applying
the
Ratio
T
est
a
Do
es
∞
X
k
=1
(
−
1)
k
−
1
k
!
converge
or
diverge?
b
Do
es
∞
X
k
=1
2
k
k
2
converge
or
diverge?
c
Do
es
∞
X
k
=1
k
converge
or
diverge?
217
Example
3.3.9
Applying
the
Ratio
T
est
Main
Ideas
When
applying
the
ratio
test,
b
e
sure
to
replace
every
k
with
k
+
1
fo
r
the
a
k
+1
term.
F
amiliarize
y
ourself
with
the
algeb
ra
rules
that
allow
you
to
simplify
ratios
of
exp
onentials
and
facto
rials.
218
Example
3.3.10
A
Strategy
fo
r
Series
T
ests
Strategy
Given
the
three
wa
ys
we
have
to
test
for
divergence
and
convergence
and
the
relative
ease
of
applying
each,
here
is
a
reasonable
app
roach
to
testing
a
series.
Check
lim
n
→∞
a
n
by
dominance
Compute
a
n
+1
a
n
Compute
lim
n
→∞
a
n
+1
a
n
P
a
n
converges
P
a
n
diverges
Inconclusive,
lo
ok
up
another
test
not
zero
zero
hard
to
tell
constant
|
r
|
≥
1
constant
|
r
|
<
1
not
constant
<
1
>
1
=
1
hard
to
tell
219
Example
3.3.10
A
Strategy
for
Series
T
ests
Let’s
apply
our
strategy
to
see
what
we
can
tell
ab
out
∞
X
n
=1
1
n
2
.
220
Section
3.3
Summa
ry
Questions
Q1
What
is
the
difference
b
et
ween
a
sequence
and
a
series?
Q2
Ho
w
do
we
evaluate
a
series?
Q3
What
is
a
geometric
series.
How
do
w
e
evaluate
one?
Q4
What
do
es
it
mean
to
say
that
a
series
test
is
inconclusive?
Q5
Ho
w
do
each
of
the
follo
wing
factors
b
ehave
in
the
ratio
a
k
+1
a
k
?
a
k
p
(
p
a
constant)
b
c
k
(
c
a
constant)
c
k
!
Q6
Ho
w
w
ould
the
ratio
test
apply
to
a
geometric
series
X
a
r
k
−
1
?
221
Section
3.3
Q52
Supp
ose
that
a
discrete
random
variable
X
has
distribution
function
f
X
(
x
)
=
(
1
x
−
1
x
+1
if
x
is
a
p
ositive
integer
0
otherwise
a
V
erify
that
f
X
(
x
)
is
a
valid
p
robability
distribution
function.
b
Compute
P
(3
≤
X
≤
5).
c
Explain
why
y
ou
can’t
compute
E
[
X
].
222
Section
3.4
P
o
w
er
Series
Goals:
1
Use
series
tests
to
determine
for
what
values
of
x
a
p
ow
er
series
converges.
2
Identify
the
radius
of
convergence
of
a
p
ow
er
series.
3
Recognize
functions
that
can
b
e
rewritten
as
a
p
o
wer
series.
Question
3.4.1
What
Is
a
P
o
w
er
Series?
So
far
we
have
studied
infinite
series
of
numbers.
If
instead
of
just
numb
ers,
our
terms
include
va
riables,
then
we’ve
created
a
function.
Plugging
in
different
values
fo
r
the
variable
gives
us
a
different
series
of
numb
ers.
Example
The
expression
1
+
x
+
x
2
+
x
3
+
·
·
·
b
ecomes
1
+
2
+
4
+
8
+
·
·
·
when
we
evaluate
it
at
x
=
2.
It
b
ecomes
1
−
1
3
+
1
9
−
1
27
+
·
·
·
when
we
evaluate
it
at
x
=
−
1
3
.
224
Question
3.4.1
What
Is
a
Po
wer
Series?
Definition
An
infinite
series
of
the
form
∞
X
k
=0
c
k
(
x
−
a
)
k
is
called
a
p
o
w
er
series
centered
at
a
.
It
is
a
function
of
x
whose
domain
is
all
values
of
x
that
mak
e
the
series
converge.
F
or
the
purp
oses
of
this
definition,
we
define
x
0
=
1
even
when
x
=
0.
225
Example
3.4.2
A
Geometric
Series
as
a
P
o
w
er
Series
Use
the
geometric
series
fo
rmula
to
write
f
(
x
)
=
1
1
−
x
as
a
p
o
wer
series
and
find
its
domain.
226
Example
3.4.3
The
Domain
of
a
P
o
w
er
Series
What
is
the
domain
of
∞
X
k
=1
k
2
4
k
(
x
−
5)
k
?
227
Example
3.4.3
The
Domain
of
a
Po
wer
Series
Main
Idea
The
ratio
test
is
usually
successful
in
finding
where
a
p
o
wer
series
converges.
Generally
it
is
inconclusive
at
only
tw
o
p
oints.
W
e
will
not
alw
ays
have
a
test
that
can
tell
us
whether
the
series
converges
at
these
p
oints.
228
Example
3.4.3
The
Domain
of
a
Po
wer
Series
Theo
rem
Given
a
p
o
wer
series
∞
X
k
=0
c
k
(
x
−
a
)
k
centered
at
a
,
one
of
the
following
is
true.
1
The
series
converges
only
when
x
=
a
.
2
The
series
converges
when
x
is
any
real
numb
er.
3
There
is
a
radius
of
convergence
R
such
that
a
The
series
converges
when
|
x
−
a
|
<
R
,
and
b
The
series
diverges
when
|
x
−
a
|
>
R
.
Click to Load Applet
Figure:
The
domain
|
x
−
a
|
<
R
of
a
p
o
wer
series.
229
Question
3.4.4
Can
W
e
Integrate
o
r
Differentiate
a
P
o
w
er
Series?
When
f
(
x
)
is
a
p
olynomial,
w
e
can
find
the
derivative
and
anti-derivative
of
f
(
x
)
b
y
computing
the
(anti-)derivative
of
each
term.
The
follo
wing
theo
rem
sa
ys
that
w
e
can
do
this
for
a
p
ow
er
series
to
o.
Theo
rem
If
f
(
x
)
is
the
pow
er
series
∞
X
k
=0
c
k
(
x
−
a
)
k
and
f
(
x
)
has
radius
of
convergence
R
>
0
then
f
(
x
)
is
differentiable
and
continuous
on
the
interval
(
a
−
R
,
a
+
R
),
and
1
f
′
(
x
)
=
∞
X
k
=1
k
c
k
(
x
−
a
)
k
−
1
2
Z
f
(
x
)
dx
=
C
+
∞
X
k
=0
c
k
(
x
−
a
)
k
+1
k
+
1
Both
of
these
functions
also
have
radius
of
convergence
R
.
230
Question
3.4.4
Can
We
Integrate
or
Differentiate
a
Po
wer
Series?
Example
W
e
have
seen
that
1
1
−
x
=
∞
X
k
=0
x
k
on
the
interval
(
−
1
,
1).
From
that
we
can
compute:
d
dx
∞
X
k
=0
x
k
=
∞
X
k
=1
kx
k
−
1
Z
∞
X
k
=0
x
k
dx
=
∞
X
k
=0
x
k
+1
k
+
1
+
c
Both
have
domain
(
−
1
,
1).
231
Section
3.4
Summa
ry
Questions
Q1
What
is
the
difference
b
et
ween
a
p
olynomial
and
a
p
o
wer
series?
Q2
What
test
is
useful
fo
r
establishing
the
domain
of
a
p
ow
er
series?
What
form
can
this
domain
have?
Q3
Ho
w
can
we
integrate
or
differentiate
a
p
ow
er
series?
Q4
Ho
w
does
differentiation
affect
the
radius
of
convergence
of
a
p
o
wer
series?
232
Section
3.4
Q18
Supp
ose
you
are
told
that
a
given
p
ow
er
series
p
(
x
)
centered
at
x
=
a
converges
at
x
=
−
4
and
diverges
at
x
=
−
7.
a
If
a
=
1,
what
can
you
say
ab
out
the
domain
of
p
(
x
)?
b
What
are
all
of
the
the
p
ossible
values
of
a
?
Explain
y
our
reasoning
(b
riefly).
233
Section
3.5
T
a
ylo
r
Series
Goals:
1
Use
a
combination
of
p
o
wer
series
and
algeb
ra
to
wo
rk
with
functions.
2
Integrate
and
differentiate
p
ow
er
series.
Question
3.5.1
What
Is
a
T
a
ylo
r
Series?
Definition
The
T
aylo
r
series
of
f
(
x
)
at
x
=
a
is
T
(
x
)
=
∞
X
k
=0
f
(
k
)
(
a
)
k
!
(
x
−
a
)
k
.
The
T
aylo
r
series’s
pa
rtial
sums
s
n
a
re
the
T
aylo
r
p
olynomials
T
n
(
x
)
of
f
at
x
=
a
.
Rema
rk
Several
mathematicians
contributed
to
the
discovery
of
T
aylo
r
series.
T
a
ylor
series
centered
at
x
=
0
were
p
opularized
by
Colin
Maclaurin,
and
so
are
often
called
Maclaurin
series
.
235
Question
3.5.1
What
Is
a
T
a
ylor
Series?
Limitations
of
T
a
ylor
Series
T
a
ylor
p
olynomials
were
designed
to
appro
ximate
f
(
x
).
We
might
hop
e
that
T
(
x
)
w
ould
be
the
p
erfect
app
roximation,
that
T
(
x
)
and
f
(
x
)
are
equal.
Unfortunately
,
there
are
obstacles
to
this.
The
T
aylo
r
series
might
not
converge
for
all
x
.
The
T
aylo
r
p
olynomials
might
not
app
roximate
f
(
x
)
very
well
at
all.
Recall
our
example
f
(
x
)
=
(
0
if
x
≤
0
e
−
1
x
if
x
>
0
F
or
this
function
T
(
x
)
=
0.
236
Example
3.5.2
W
riting
a
T
a
ylo
r
series
Let
f
(
x
)
=
e
x
a
Find
the
T
a
ylor
series
for
f
(
x
)
centered
at
x
=
0.
b
On
what
interval
do
es
it
converge?
237
Synthesis
3.5.3
Is
a
T
a
ylor
Series
Equal
to
the
Function
it
App
roximates?
Let
f
(
x
)
=
ln
x
a
Find
a
pattern
in
the
derivatives
and
write
a
general
exp
ression
fo
r
the
k
th
derivative:
f
(
k
)
(
x
).
b
Use
your
answer
to
a
to
write
exp
ressions
fo
r
the
T
a
ylor
p
olynomials
T
n
(
x
)
and
the
T
aylo
r
series
T
(
x
)
of
ln
x
centered
at
1.
Simplify
the
co
efficients
if
p
ossible.
c
What
do
es
the
ratio
test
tell
y
ou
about
where
T
(
x
)
converges?
d
If
w
e
wanted
to
apply
T
aylo
r’s
inequalit
y
to
T
n
(
x
),
we
would
need
to
kno
w
where
the
derivative
is
largest
(in
absolute
value).
Where
is
the
(
n
+
1)th
derivative
la
rgest
on
the
interval
[
x
,
1]?
(Here
0
<
x
<
1).
e
Where
is
the
(
n
+
1)th
derivative
largest
on
the
interval
[1
,
x
]?
(Here
x
>
1).
f
What
do
es
T
a
ylor’s
inequalit
y
sa
y
ab
out
where
R
n
(
x
)
→
0
as
n
→
∞
?
g
What
do
es
our
answer
to
the
previous
question
tell
us
about
T
(
x
)?
238
Synthesis
3.5.3
Is
a
T
aylor
Series
Equal
to
the
Function
it
Approximates?
a
Find
a
pattern
in
the
derivatives
and
write
a
general
exp
ression
fo
r
the
k
th
derivative:
f
(
k
)
(
x
).
238
Synthesis
3.5.3
Is
a
T
aylor
Series
Equal
to
the
Function
it
Approximates?
b
Use
your
answer
to
a
to
write
exp
ressions
fo
r
the
T
a
ylor
p
olynomials
T
n
(
x
)
and
the
T
aylo
r
series
T
(
x
)
of
ln
x
centered
at
1.
Simplify
the
co
efficients
if
p
ossible.
238
Synthesis
3.5.3
Is
a
T
aylor
Series
Equal
to
the
Function
it
Approximates?
c
What
do
es
the
ratio
test
tell
y
ou
about
where
T
(
x
)
converges?
238
Synthesis
3.5.3
Is
a
T
aylor
Series
Equal
to
the
Function
it
Approximates?
d
If
w
e
wanted
to
apply
T
aylo
r’s
inequalit
y
to
T
n
(
x
),
we
would
need
to
kno
w
where
the
derivative
is
largest
(in
absolute
value).
Where
is
the
(
n
+
1)th
derivative
la
rgest
on
the
interval
[
x
,
1]?
(Here
0
<
x
<
1).
238
Synthesis
3.5.3
Is
a
T
aylor
Series
Equal
to
the
Function
it
Approximates?
e
Where
is
the
(
n
+
1)th
derivative
largest
on
the
interval
[1
,
x
]?
(Here
x
>
1).
238
Synthesis
3.5.3
Is
a
T
aylor
Series
Equal
to
the
Function
it
Approximates?
f
What
do
es
T
a
ylor’s
inequalit
y
sa
y
ab
out
where
R
n
(
x
)
→
0
as
n
→
∞
?
238
Synthesis
3.5.3
Is
a
T
aylor
Series
Equal
to
the
Function
it
Approximates?
g
What
do
es
our
answer
to
the
previous
question
tell
us
about
T
(
x
)?
238
Synthesis
3.5.3
Is
a
T
aylor
Series
Equal
to
the
Function
it
Approximates?
Click to Load Applet
Figure:
The
T
a
ylor
p
olynomials
app
roach
ln
x
only
on
(0
,
2].
239
Example
3.5.4
Mixing
T
a
ylo
r
Series
and
Algeb
ra
Let
f
(
x
)
=
x
2
sin
x
.
Compute
a
T
aylo
r
series
for
f
(
x
)
centered
at
x
=
0.
240
Example
3.5.4
Mixing
T
aylor
Series
and
Algebra
Main
Idea
When
constructing
a
T
aylo
r
series
fo
r
f
(
x
)
=
x
k
g
(
x
)
centered
at
0,
construct
the
T
a
ylor
series
of
g
(
x
),
and
then
distribute
the
x
k
.
241
Example
3.5.5
Integrating
a
T
a
ylo
r
Series
Let
f
(
x
)
=
e
x
2
.
a
W
rite
a
T
aylo
r
polynomial
T
4
(
x
)
for
f
(
x
)
at
x
=
0.
b
Find
a
b
etter
wa
y
to
p
ro
duce
the
T
aylo
r
series
for
f
(
x
).
c
Compute
a
T
a
ylor
series
for
Z
e
x
2
dx
.
242
Example
3.5.5
Integrating
a
T
aylor
Series
a
W
rite
a
T
aylo
r
polynomial
T
4
(
x
)
for
f
(
x
)
at
x
=
0.
242
Example
3.5.5
Integrating
a
T
aylor
Series
b
Find
a
b
etter
wa
y
to
p
ro
duce
the
T
aylo
r
series
for
f
(
x
).
242
Example
3.5.5
Integrating
a
T
aylor
Series
b
Compute
a
T
a
ylor
series
for
Z
e
x
2
dx
.
242
Example
3.5.5
Integrating
a
T
aylor
Series
Click to Load Applet
Figure:
The
graph
of
e
x
2
,
R
e
x
2
dx
,
and
the
pa
rtial
sums
of
its
T
aylo
r
series.
Main
Ideas
Comp
ositions
of
functions
can
b
e
comp
osed
through
T
aylo
r
series.
T
a
ylor
series
allow
us
to
integrate
functions
that
a
re
otherwise
imp
ossible
to
integrate.
243
Application
3.5.6
Euler’s
F
o
rmula
Recall
i
is
an
imaginary
numb
er
that
satisfies
i
2
=
−
1.
a
Find
an
exp
ression
fo
r
f
(
x
)
=
e
ix
.
b
W
rite
y
our
answ
er
in
terms
of
the
T
aylo
r
series
for
sin
x
and
cos
x
.
c
W
rite
tw
o
different
exp
ressions
fo
r
e
i
2
x
.
How
is
this
equation
useful?
244
Section
3.5
Summa
ry
Questions
Q1
Ho
w
can
we
b
e
sure
that
a
T
a
ylor
series
converges
to
the
function
it
is
appro
ximating?
Q2
What
is
the
domain
of
a
T
a
ylor
series?
Q3
Ho
w
can
we
p
ro
duce
the
T
aylo
r
series
for
x
n
f
(
x
)
o
r
f
(
x
n
)?
Where
do
es
the
center
need
to
b
e
fo
r
the
result
to
b
e
a
T
aylo
r
series?
Q4
What
is
a
Maclaurin
series?
245
Section
3.5
Q16
Supp
ose
for
a
function
f
we
a
re
able
to
place
a
bound
of
k
!
3
k
on
the
k
th
derivative
of
f
over
any
interval.
Fo
r
what
values
of
x
can
w
e
conclude
that
T
(
x
),
the
T
aylo
r
series
ce
ntered
at
2,
is
equal
to
f
(
x
)?
246
Section
3.5
Q18
F
or
a
general
function
f
and
its
T
aylo
r
p
olynomials
and
series,
ho
w
a
re
the
following
sets
of
points
related?
Do
es
every
number
belonging
to
one
of
these
sets
b
elong
to
one
of
the
others?
The
set
of
numb
ers
x
where
T
(
x
)
converges.
The
set
of
numb
ers
x
where
|
R
n
(
x
)
|
→
0
as
n
→
∞
.
The
set
of
numb
ers
where
f
(
x
)
=
T
(
x
).
247
Section
3.5
Q26
W
rite
the
T
aylo
r
series
for
f
(
x
)
=
1
x
centered
at
1.
Verify
that
one
of
its
antiderivatives
is
a
T
aylo
r
series
fo
r
ln
x
.
248
Section
3.5
Q34
Supp
ose
we
have
a
function
f
(
x
)
and
tw
o
different
numb
ers
a
and
b
.
Supp
ose
further
that
the
T
a
ylor
series
for
f
(
x
)
centered
at
a
is
equal
to
the
T
aylo
r
series
for
f
(
x
)
centered
at
b
.
What
can
y
ou
sa
y
ab
out
the
domain
of
this
T
aylo
r
series?
249
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