Notes on Series
Definitions
: Let
{
}
1
n
n
a
∞
=
be a sequence. Define a new sequence
{
}
1
n
n
s
∞
=
, called the sequence
of partial sums
for the sequence
{
}
1
n
n
a
∞
=
, by the following equations:
1
1
s
a
=
,
1
n
n
n
s
s
a

=
+
for
2
n
≥
.
This new sequence of partial sums is called the series
of
n
a
s, and it is denoted by
1
n
n
a
∞
=
∑
.
For each
1
n
≥
,
n
s
is called the
th
n
partial sum
for the series
1
n
n
a
∞
=
∑
. The number
n
a
is called
the
th
n
term
of the series (as well as the
th
n
term of the sequence).
There are several observations to keep in mind when dealing with series. These are listed
below:
Notice that the sequence of partial sums and the series of
n
a
are mathematically
the same thing.
The term
n
s
is the
th
n
term of the sequence of partial sums
{
}
1
n
n
s
∞
=
. If
{
}
1
n
n
s
∞
=
is the
same thing as the series
1
n
n
a
∞
=
∑
, one might ask why is
n
s
not called the
th
n
term of the
series? The reason is, as we shall see later, that it is necessary to distinguish
between
n
a
and
n
s
without specific reference to the sequence
{
}
1
n
n
a
∞
=
.
For
2
n
≥
,
1
n
n
n
s
s
a

=
+
. This means
1
1
s
a
=
2
1
2
1
2
s
s
a
a
a
=
+
=
+
3
2
3
1
2
3
s
s
a
a
a
a
=
+
=
+
+
4
3
4
1
2
3
4
s
s
a
a
a
a
a
=
+
=
+
+
+
M
1
1
2
3
n
n
n
n
s
s
a
a
a
a
a

=
+
=
+
+
+
+
L
So we often write
1
2
3
1
n
n
n
a
a
a
a
a
∞
=
=
+
+
+
+
+
∑
L
L
Sometimes the index for the sequence
{
}
1
n
n
a
∞
=
does not begin with the number 1. In
such cases the corresponding series is also indexed accordingly. For example, if
the sequence is
{
}
0
n
n
a
∞
=
, then the series is
0
1
2
0
n
n
n
a
a
a
a
a
∞
=
=
+
+
+
+
+
∑
L
L
; while if
the sequence is
{
}
3
n
n
a
∞
=
, then the series is
3
4
5
3
n
n
n
a
a
a
a
a
∞
=
=
+
+
+
+
+
∑
L
L
.
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When the index for the sequence
{
}
n
a
does not matter in the discussion, we write
n
a
∑
, or, in the context of the sequence of partial sums, we write
{
}
n
s
.
Examples
: Below are some examples that illustrate the difference between a sequence
and the corresponding series and the sequence of partial sums.
1)
Consider the constant sequence
{
}
1
n
a
∞
=
, where
a
is a constant.
Observe that
{
}
{
}
1
, , ,
, ,
n
a
a a a
a
∞
=
=
K K
. The corresponding series is
1
n
a
a
a
a
a
∞
=
=
+ + +
+ +
∑
L
L
.
Calculating the sequence of partial sums:
1
1
2
1
2
1
2
3
2
3
1
2
3
4
3
4
1
2
3
4
1
1
2
3
2
3
4
n
n
n
n
n times
s
a
a
s
s
a
a
a
a
a
a
s
s
a
a
a
a
a
a
a
a
s
s
a
a
a
a
a
a
a
a
a
a
s
s
a
a
a
a
a
a
a
a
a
na


=
=
=
+
=
+
=
+
=
=
+
=
+
+
=
+ +
=
=
+
=
+
+
+
=
+ + +
=
=
+
=
+
+
+
+
=
+ + +
+
=
M
L
L
1 4 42 4 43
So the sequence of partial sums is
{
}
{
}
1
,2 ,3 ,
,
,
n
an
a
a
a
an
∞
=
=
K
K
.
2)
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 Spring '11
 BUEKMAN
 Equations, Geometric Series

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