Sequence and Summation Notation
Sequence and Summation Notation
A sequence is an ordered set of numbers 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,…
a,b,c,d,e,…,z
For example, the numbers from 1 to 10 are a finite sequence: . A positive even number can be represented by , where is a positive integer, giving the infinite sequence .
1,2,3,4,5,6,7,8,9,10
2k
k
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: ,,…. Such an integer is called an index.
a
1
a
2
a
3
More compactly, sequence notation is used: means ,,…,. If the number of terms is infinite, the sequence ends with "…", like this: =,,,….
n
{}
a
k
k=1
a
1
a
2
a
3
a
n
∞
{}
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 , which stands for +++⋯+.
n
∑
k=1
a
k
a
1
a
2
a
3
a
n
For infinitely many terms, the notation is , which stands for +++⋯.
∞
∑
k=1
a
k
a
1
a
2
a
3