Exam 2: Cheat Sheet"
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Exam 2: Cheat Sheet"
Z
Z
n
7.
sec
= tan
x
6
+1
1.
=
+ 1
= 1
n
2
x dx
x
x
dx
n
n
Z
Z
1
8.
csc
= cot
j
j
2.
= ln
2
x dx
x
dx
x
x
Z
Z
9.
sec
tan
= sec
3.
=
x
x
x
x dx
x
e
dx
e
Z
Z
x
10.
csc
cot
= csc
a
4.
=
x
x
x dx
x
a
dx
ln
a
Z
= 1
Z
dx
x
11.
tan
5.
sin
= cos
1
+
x dx
x
x
a
a
a
2
2
Z
Z
p
dx
x
12.
= sin
6.
cos
= sin
1
x dx
x
a
a
x
2
2
limit
f
g
j
j
13. A sequence
has
if for every
0 there is an integer
such that
whenever
a
L
N
a
L
n
n
.
n
N
1
14. lim
=
means for every positive number
there is an integer
such that
whenever
!1
a
M
N
a
M
n
n
n
.
n
N
Squeeze Theorem.
15.
If
for
and lim
= lim
=
, then lim
=
!1
!1
!1
a
b
c
n
n
a
c
L
b
n
n
n
n
n
n
n
n
n
0
.
L
Theorem.
j
j
16.
If lim
= 0, then lim
= 0.
!1
!1
a
a
n
n
n
n
f
g
17.
is convergent if 1
1 and divergent for all other values of
.
n
r
r
r
Theorem.
18.
Every bounded, monotonic sequence is convergent.
1
1
P
P
j
j
j
j
19. The geometric series
is convergent if
1 and its sum is
=
. If
1,
n
n
a
ar
1
r
ar
1
r
n
n
r
the geometric series is divergent.
=1
=1
1
1
P
Test for Divergence.
6
20.
If lim
does not exist or if lim
= 0, then the series
is
!1
!1
a
a
a
n
n
n
n
n
n
divergent.
=1
The Integral Test.
1
21.
Suppose
is a continuous, positive, decreasing function on 1
and let
f
;
1
1
R
P
=
. The the series
is convergent if and only if the improper integral
is
a
f
n
a
f
x
dx
n
n
n
convergent.
=1
1
Remainder Estimate for the Integral Test.
22.
If
converges by the Integral Test and
=
,
a
R
s
s
n
n
n
then
1
1
Z
Z
f
x
dx
R
f
x
dx
n
n
n
+1
P
P
The Comparison Test.
23.
Suppose that
and
are series with positive terms.
a
b
n
n
P
P
a If
is convergent and
for all
, then
is convergent.
b
a
b
n
a
n
n
n
n
P
P
b If
is divergent and
for all
, then
is divergent.
b
a
b
n
a
n
n
n
n
P
P
Limit Comparison Test.
24.
Suppose that
and
are series with positive terms.
a
b
n
n
a If lim
=
0, then either both series converge or both diverge.
a n
!1
c
n
b n
P
P
b If lim
= 0 and
converges, then
also converges.
a n
!1
b
a
n
n
n
b n
P
P
1
c If lim
=
and
diverges, then
also diverges.
a n
!1
b
a
n
n
n
b n
9
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