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Math 2224 Multivariable Calculus – Sec. 8.3: The Integral Test
I.
Nondecreasing Partial Sums
A.
Corollary of Theorem 6
A series
a
n
n
=
1
!
"
of nonnegative terms converges iff its partial sums are bounded from
above.
B.
Example
The harmonic series
1
n
=
1
+
1
2
+
1
3
+
...
+
1
n
+
...
n
=
1
!
"
diverges because there is no upper
bound for its partial sum.
Group the terms of the series as follows:
The sum of the
2
n
terms ending in
1
2
n
+
1
is greater than
2
n
2
n
+
1
=
1
2
. If
n
=
2
k
the partial
sum
s
n
is greater than
k
2
. The harmonic series diverges.
II.
Integral Test
A. Introduction
Consider the series
1
n
2
=
1
+
1
4
+
1
9
+
1
16
+
...
+
1
n
2
+
...
n
=
1
!
"
. To determine if the series
converges, compare it with
1
x
2
1
!
"
dx
. We can interpret this as the area of rectangles
under the curve
f
(
x
)
=
1
x
2
.
s
n
=
1
1
2
+
1
2
2
+
1
3
2
+
1
4
2
+
...
+
1
n
2
=
f
(1)
+
f
(2)
+
f
(3)
+
f
(4)
+
...
+
f
(
n
)
< f
(1)
+
1
x
2
dx
1
n
!
<
1
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This note was uploaded on 04/05/2008 for the course MATH 2224 taught by Professor Mecothren during the Spring '03 term at Virginia Tech.
 Spring '03
 MECothren
 Harmonic Series, Multivariable Calculus

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