A sequence is a special kind of function whose domain is the positive integers. The range of a sequence is the collection of terms that make up the sequence. Just as the word sequence implies, the order of the terms in a sequence is important. The first term of a sequence, for example, is found by taking the value of the function at 1 ; the second term is the value of the function at 2 , and so on. Consider the sequence f (x) = x . The terms of the sequence, denoted a 1, a 2, a 3, , a n are 1, 2, 3, , n . When working with sequences, instead of using function notation to express the formula of the function, a formula is the following form is used: a n = n . This is the same sequence as above, but the conventional n is used to denote an integer, since only integers are in the domain of sequences. Two important categories of sequences are arithmetic sequences, and geometric sequences. Both are examples of a recursive sequence--a sequence in which each term (besides the first) depends on the previous term. Both of these types of sequences will be discussed.
When the terms of a sequence are summed, the result is called a series. Some series increase without bound as n increases, but others approach a limit. Both types of series will be studied in the following sections. There are also certain formulas for calculating the limits of series that we'll learn. The study of series is an important part of calculus, and it all starts with sequences.