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7.2 Numerical Series and Sequences
103
Shifting one unit to the right, the region under the graph is contained in the rectangle, so
0
≤
A
n
≤
f
(
n
)
,
for
n
≥
N.
This gives
N
+
n
X
k
=
N
+1
f
(
k
)
≤
Z
N
+
n
N
+1
f
(
x
)
dx
≤
N
+
n

1
X
k
=
N
f
(
k
)
≤
Z
N
+
n

1
N
f
(
x
)
dx
and so the two sequences converge or diverge together.
Theorem 7.2.18
(pseries)
.
∑
1
n
p
converges iﬀ
p >
1
.
Proof.
For
p
≥
0,
1
n
p
is decreasing, so apply the integral test to
Z
∞
1
1
x
p
dx
= lim
r
→∞
Z
r
1
1
x
p
dx
=
lim
r
→∞
r
1

p

1
1

p
, p
6
= 1
,
lim
r
→∞
log
r,
p
= 1
.
For
p
= 1, log
r
→ ∞
and both diverge.
For
p >
1,
r
1

p
→
0, so both are ﬁnite.
For 0
≤
p <
1,
r
1

p
→ ∞
, so both diverge.
Finally, consider
p <
0 and put
q
=

p >
0. Then
∑
n
q
diverges by
n
th
term test.
Theorem 7.2.19
(Asymptotic comparison test)
.
If
lim

a
n


b
n

=
L
, where
L
∈
(0
,
∞
)
then
X

a
n

converges
⇐⇒
X

b
n

converges
.
Proof.
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 Fall '11
 Wong

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