Sequences are to calculus what at calculator is to a scientist. There are many ways to introduce sequences. Here we will follow a somewhat unorthodox way. Indeed, consider a scientist doing an experiment; he is collecting data, let us say, every day. So, put to be the data collected the first day, be the data collected the second day, and so on…. and is the data collected after n days. Clearly, we are generating a set of numbers with a very special characteristic: there is an order on the number, that is, we naturally have the first number, the second number, and so on…. A sequence is by definition a set of real numbers with this natural order. We wil use the notation
to describe the sequence of numbers where is the nth number.
Definition (Range): Consider the sequence . The set
is called the range of the sequence.
Of course, in the range there is no order. For example, consider the sequence . Its range is the set . It has two elements. The sequence itself is alternating between 1 and -1.