Chapter 2 Expanded Notes

Chapter 2 Expanded Notes - 2 Sequences and Series In this...

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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 | < ....
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Chapter 2 Expanded Notes - 2 Sequences and Series In this...

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