WOLFRAM|DEMONSTRATIONS PROJECT

Sequence and Summation Notation

​
n
1
2
3
4
5
6
7
8
9
∞
terms
a
k
k
2
k
k!
k
2
1
k
1
2
k
1
k!
1
k
2
sequence
{
a
k
} =
a
1
,
a
2
,
a
3
series
3
∑
k = 1
a
k
=
a
1
+
a
2
+
a
3
A sequence is an ordered set of numbers
a,b,c,d,e,…
that may have a finite or infinite number of terms. If the sequence is finite, the last term is shown, like
a,b,c,d,e,…,z
.
For example, the numbers from 1 to 10 are a finite sequence:
1,2,3,4,5,6,7,8,9,10
. A positive even number can be represented by
2k
, where
k
is a positive integer, giving the infinite sequence
2,4,6,8,…
.
The character "…" (called an ellipsis) means "keep going as before."
To avoid using up many different letters, often the same letter is used with a whole number to its right and below (called a subscript), like this:
a
1
a
2
,
a
3
,…
. Such an integer is called an index.
More compactly, sequence notation is used:
n
{
a
k
}
k=1
means
a
1
a
2
,
a
3
,…,
a
n
. If the number of terms is infinite, the sequence ends with "…", like this:
∞
{
a
k
}
k=1
=
a
1
,
a
2
,
a
3
,…
.
A series is the sum of a sequence, for example,
1+2+3+4+5+6+7+8+9+10
.
Like a sequence, the number of terms in a series may be finite or infinite.
The notation for a series with finitely many terms is
n
∑
k=1
a
k
, which stands for
a
1
+
a
2
+
a
3
+⋯+
a
n
.
For infinitely many terms, the notation is
∞
∑
k=1
a
k
, which stands for
a
1
+
a
2
+
a
3
+⋯
.