This preview shows pages 1–3. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: 2 Sequences and Series In this chapter we will study two related questions. Given an infinite collec tion X of numbers, which can be taken to be rational, real or complex, the first question is to know if there is a limit point to which the elements of X congregate (converge). The second question is to know if one can add up all the numbers in X . Even though we will be mainly thinking of Q , R or C , many things will go through for numbers in any field K with an absolute value  .  satisfying   = 0,  x  > 0 if x = 0,  xy  =  x   y  , and the triangle inequality  x + y   x  +  y  . A necessary thing to hold for convergence is the Cauchy condition , which is also sucient if we work with R or C , but not Q . What we mean by this remark is that a Cauchy sequence of rational numbers will have a limit L in R , but L need not be rational. One can enlarge Q to have this criterion work, i.e., adjoin to Q limits of Cauchy sequences of rational numbers, in which case one gets yet another construction of R . This aspect is discussed in section 2.3, which one can skip at a first reading. The last three sections deal with the second question, and we will derive various tests for absolute convergence . Sometimes, however, a series might converge though not absolutely. A basic example of this phenomenon is given by the Leibniz series 1 1 3 + 1 5 1 7 + 1 9 1 11 + . . . = 4 . But the series on the left is not absolutely convergent, i.e., the series 1 + 1 3 + 1 5 + . . . diverges. 2.1 Convergence of sequences Let K denote either Q or R or C . When we say a number , or a scalar , we will mean an element of K . But by a positive number we will mean a positive real number. (One could restrict to positive rational numbers instead, and it will work just as well to use this subset.) By a sequence , or more properly an infinite sequence , we will mean a collection of numbers a 1 , a 2 , a 3 , . . . , a n , a n +1 , . . . , 1 which is indexed by the set N of natural numbers. We will often denote it simply as { a n } . A simple example to keep in mind is given by a n = 1 n , which appears to decrease towards zero as n gets larger and larger. In this case we would like to have 0 declared as the limit of the sequence . It should be noted that if we are given a collection of numbers indexed by any countably infinite set, we can reindex it by N and apply all our results below to it. For example, given M > 0 and { c n  n > M } , we can renumber it by putting a n = c n + M and get a sequence indexed by N . We will say that a sequence { a n } converges to a limit L in K iff for every positive number there exists an index N such that (2 . 1 . 1) n > N =  L a n  < ....
View
Full
Document
 Fall '09
 E.Ryckman
 Calculus, Sequences And Series

Click to edit the document details