Introduction to Sequences

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 tex2html_wrap_inline16 to be the data collected the first day, tex2html_wrap_inline18 be the data collected the second day, and so on.... and tex2html_wrap_inline20 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

displaymath22,

to describe the sequence of numbers where tex2html_wrap_inline20 is the nth number.

Definition (Range): Consider the sequence tex2html_wrap_inline26 . The set

displaymath28

is called the range of the sequence.

Of course, in the range there is no order. For example, consider the sequence tex2html_wrap_inline30 . Its range is the set tex2html_wrap_inline32 . It has two elements. The sequence itself is alternating between 1 and -1.

For more on sequences, click Here.

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Mohamed A. Khamsi
Tue Dec 3 17:39:00 MST 1996

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