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7.2 Numerical Series and Sequences
97
7.2 Numerical Series and Sequences
7.2.1 Convergence and absolute convergence
Deﬁnition 7.2.1.
An (inﬁnite)
series
is a sum of a sequence
{
a
k
}
:
∞
X
k
=0
a
k
=
a
0
+
a
1
+
a
2
+
....
To make it clear that the terms of the sequence
{
a
k
}
are added in order, deﬁne
∞
X
k
=0
a
k
= lim
n
→∞
n
X
k
=0
a
k
= lim
s
n
,
where
s
n
:=
∑
n
k
=0
a
k
. The series
∑
a
k
converges or diverges
as the sequence
s
n
does.
Thus, a series is a any sequence which can be written in a certain simple recursive
form:
s
n
=
s
n

1
+
f
(
n
)
.
Example 7.2.1.
Geometric series: 1 +
r
+
r
2
+
···
=
∑
∞
k
=0
r
k
is the limit of
s
n
= 1 +
r
+
r
2
+
···
+
r
n
=
s
n

1
+
r
n
.
Example 7.2.2.
Harmonic series: 1 +
1
2
+
1
3
+
···
=
∑
∞
k
=1
1
k
is the limit of
s
n
= 1 +
1
2
+
1
3
+
···
+
1
n
=
s
n

1
+
1
n
.
Deﬁnition 7.2.2.
A
telescoping series
is one that can be written in the form
∞
X
k
=0
(
a
k
+1

a
k
)
.
A telescoping series has partial sums
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 Fall '11
 Wong

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