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Notes on Series  Continued
Convergent series may be thought of as finite sums as they ‘sum’ to a finite number.
Therefore we can have an arithmetic associated with them. Divergent series though do
not have arithmetical properties. Convergent series may be multiplied by a constant
number, added together, subtracted from each other, multiplied together and divided. In
this course we only consider the first three arithmetic operations though. The theorem
below provides the relevant statements.
Theorem
: A) Suppose
n
a
∑
and
n
b
∑
are convergent series and
c
is a constant. Then the
series
n
ca
∑
and
(
29
n
n
a
b
±
∑
are all convergent series, and
n
n
ca
c
a
=
∑
∑
(
29
n
n
n
n
a
b
a
b
±
=
±
∑
∑
∑
B) Suppose
n
a
∑
diverges and
c
is a constant. Then
n
ca
∑
diverges for
0
c
≠
, and, if
n
b
∑
is a convergent series, then
(
29
n
n
a
b
±
∑
diverges.
Examples
: Below are some examples of how this theorem may be used to determine
convergence/divergence of certain series.
1)
Show the series
1
1
1
1
2
2
3
n
n
n
∞
+

=

∑
converges and find the sum of the series.
Consider
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This note was uploaded on 05/06/2011 for the course MATH 256 taught by Professor Buekman during the Spring '11 term at Purdue UniversityWest Lafayette.
 Spring '11
 BUEKMAN
 Harmonic Series

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