TABLE OF CONTENTS The method of exhaustion and its variations shows us how to add smaller and smaller bits of area or length so that the accumulated total approaches some limiting area or length. Certain sets of real numbers may also approach some limiting value. We begin with some definitions.
A sequence is a succession of terms in a particular order and is formed according to some fixed rule. For example:
The first, of course, is the sequence of squares of the natural numbers, and the second follows from a somewhat more complicated formula, soon explained.
A series is the sum of the terms of a sequence. From the above sequences we construct the following series:
When the number of terms is fixed and countable, the sequence or series is finite. An infinite sequence has infinitely many terms, as many terms, in fact, as there are natural numbers. In these sequences, we are usually interested in what happens in the long run. We want to find out what happens to the nth term when n becomes very large. For example, in the sequence 1, 1/2, 1/3, 1/4, 1/5, , 1/ n, the terms get smaller and smaller, getting closer and closer to 0. This infinite sequence approaches zero as the limiting value. We indicate this by writing
again, where "lim" stands for limit and " n ? ?" stands for n approaches infinity or, in other words n
