Exam 2: Cheat Sheet" Page 2
ADVERTISEMENT
1
21
P
The Alternating Series Test.
25.
If the alternating series
1
satis es a
for
n
1
b
b
b
n
n
n
n
+1
all
and b lim
= 0 then the series is convergent.
=1
!1
n
b
n
n
P
Alternating Series Estimation Theorem.
26.
If
=
1
is the sum of an alternating series
n
s
1
b
n
j
j
j
j
that satis es a and b above, then
=
R
s
s
b
n
n
n
+1
P
P
absolutely convergent
conditionally convergent
j
j
27. A series
is
if
is convergent and
if
a
a
n
n
P
is convergent but not absolutely convergent.
a
n
The Ratio Test
28.
1
P
a If lim
=
1, then the series
is absolutely convergent.
a n
+1
!1
L
a
n
n
a n
n
=1
1
P
b If lim
=
1 or is in nity, then the series
is divergent.
a n
+1
!1
L
a
n
n
n
a n
=1
The Root Test
29.
p
1
P
j
j
a If lim
=
1, then the series
is absolutely convergent.
n
!1
a
L
a
n
n
n
n
p
=1
1
P
j
j
b If lim
=
1 or is in nity, then the series
is divergent.
n
!1
a
L
a
n
n
n
n
=1
1
P
Theorem.
30.
For a given power series
there are only three possibilities: i The series
n
c
x
a
n
n
converges only when
=
. ii The series converges for all
. iii There is a positive number
such
=0
x
a
x
R
j
j
j
j
that the series converges if
and diverges if
.
x
a
R
x
a
R
1
P
P
Theorem.
31.
If
has radius of convergence
0 then the function
=
n
n
c
x
a
R
f
x
c
x
a
n
n
n
is di erentiable on
+
and
=0
a
R; a
R
1
P
0
a
=
n
f
x
nc
x
a
1
n
n
=1
+1
R
1
P
n
b
=
+
, each with radius of convergence
.
x a
f
x
dx
C
c
R
n
n
n
=0
+1
Theorem.
j
j
32.
If
has a power series representation at
, that is, if
=
for
,
n
f
a
f
x
c
x
a
x
a
R
n
n
f
a
then its coe cients are of the form
!
n
Taylor's Formula.
33.
If
has
+ 1 derivatives in an interval
that contains the number
, then for
f
n
I
a
in
there is a number
strictly between
and
such that the remainder term in the Tayler series
x
I
z
x
a
can be expressed as
n
f
+1
z
=
+ 1 !
n
R
x
x
a
+1
n
n
1
1
X
34.
=
. IOC= 1
1 .
n
x
;
1
x
n
=0
1
n
X
x
1
1
35.
=
! . IOC=
.
x
e
;
n
n
=0
1
n
X
x
2
+1
1
1
36. sin
=
1
+ 1 ! . IOC=
.
n
x
;
2
n
n
=0
1
n
X
1
1
x
2
37. cos
=
1
! . IOC=
.
n
x
;
2
n
n
=0
1
X
n
x
2
+1
38. tan
=
1
+ 1 . IOC= 1
1 .
n
1
x
;
2
n
n
=0
1
X
The Binomial Series
j
j
k
39.
If
is any real number and
1, then 1 +
=
where
k
n
k
x
x
x
n
n
=0
=
for
1 and
= 1.
k
k
k
k n
k
1
+1
n
n
n
!
0
10
ADVERTISEMENT
0 votes
Related Articles
Related forms
Related Categories
Parent category: Education