Sequence and Series
A sequence is a list of numbers in a certain order. Each number in a sequence is called a term . Each term in a sequence has a position (first, second, third and so on).
For example, consider the sequence
In the sequence, each number is called a term. The number has first position, has second position, has third position and so on.
The term of a sequence is sometimes written .
Often, you can find an algebraic expression to represent the relationship between any term in a sequence and its position in the sequence.
In the above sequence, the term can be calculated using the equation .
Finite and Infinite Sequences
A sequence is finite if it has a limited number of terms and infinite if it does not.
The first of the sequence is and the last term is . Since the sequence has a last term, it is a finite sequence.
The first term of the sequence is . The "..." at the end indicates that the sequence goes on forever; it does not have a last term. It is an infinite sequence.
A series is the indicated sum of the terms of a sequence.
Finite sequence :
Related finite series :
Infinite sequence :
Related infinite series :
A series can be written in an abbreviated form by using the Greek letter (sigma), called the summation sign. For instance, to abbreviate the writing of the series first notice that the general term of the series is . The series begins with the term for and ends with the term for . Using sigma notation, you can write this series as , which is read “the sum of for values of from to .”
Finite and Infinite Series
As in the case of sequences, we can define finite series and infinite series.
A series is finite if the corresponding sequence has a limited number of terms and infinite if it does not.
The first of the corresponding sequence is and the last term is . Since the corresponding sequence has a last term, it is a finite series.
The first term of the sequence is . The "..." at the end indicates that the series goes on forever; it does not have a last term. It is an infinite series.