The basic concepts of sequences and its limits were briefly introduced previously. A more in-depth discussion of sequences and series will be presented in this section.

Sequence {1/2n} Plotted in a
Number Line

Sequence {-(-1)ⁿ} Plotted in a
Coordinate Plane

The Plot of Sequence {1/n}

Demonstration of the Limit Definition

A sequence is an infinite ordered list of numbers:

     a₁, a₂, a₃, ..., a_n

where n is positive integer and a_n is referred to as the n^th term of the sequence. It is not necessary for n to start from one. Sometimes the sequence is denoted by {a_n} or simply a_n. For example, {1, -1, 1, -1, ...} and {2, 8, 18, 32, ...} are sequences. Normally the numbers can be expressed in terms of a formula. For example, the sequence a_n = 3n for 1 ≤ n is {3, 6, 9, ...}. However, some sequences are random and they cannot be expressed using an equation. Sequences can be expressed using two graphical representations: (a) number line and (b) coordinate plane. Two examples of sequences are shown in the figures.

Very often, as n of a sequence increases, the n^th term of the sequence approaches a particular number. Take the sequence a_n = 1/n for example. As n increases, a_n approaches 0. Hence the limit for this sequence is 0. The formal definition of a limit is given as follows:

A sequence a_n has the limit L and one can write
     as or
if there is a corresponding integer N for every ε > 0 such that

      for n > N

A sequence converges and is convergent if it has a finite limit. Otherwise, the sequence diverges or is said to be divergent. The figure on the left is used to further illustrate the definition. In order for a limit of a sequence to exist (or converges), the points of the sequence must lie between the lines (L + ε) and (L - ε) for n > N.

Some other relations such as the limit laws and squeeze theorem may be useful in determining the limit of sequence are presented next. The limit laws for sequences were introduced in the previous section, and they are summarized without further discussion as follows:

If a_n and b_n are convergent sequences and c is an arbitrary constant, then

   (a) Constant Law:
        

   (b) Law of Addition:
        

   (c) Law of Subtraction:
        

   (d) Constant Multiple Law:
        

   (e) Law of Multiplication:
        

   (f) Law of Division:
         if

Another useful relation is the squeeze theorem for a sequence:

     If a_n ≤ b_n ≤ c_n for n ≥ n_o and
     , then

When one adds the terms of an infinite sequence as follows:

     a₁ + a₂ + a₃ + ... + a_n

the above expression is referred to as an infinite series, and it is denoted by . And the n^th partial sum of the series is given by .

For a series , and S_n is its n^th partial sum

If the sequence S_n is convergent and its limit exists, then the series is convergent and one can write where S is the sum of the series.

It is said that a series converges if it has a finite sum and a series is divergent when it has an infinite sum or no sum. Take the sequence a_n = 1/2ⁿ for example. As shown in the table on the left, this series converges to 1 (i.e., ).

Another important infinite series is the geometric series, and it is defined as
     
where the number r is the common ratio of two successive terms of the series.

It is defined that the geometric series is convergent if and it is divergent if . For example, the geometric series is convergent since r = -1/3. And the geometric series is equal to -1 as shown in the figure.

Similar to the discussion of the sequences, the following laws are applied to the series:

If Σa_n and Σb_n are convergent series and c is an arbitrary constant, then

   (a) Constant Law:
        

   (b) Law of Addition:
        

   (c) Law of Subtraction: