# 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