CHAPTER 5

ELEMENTS OF CALCULUS

5.1 Sequences and Limits

A sequence of real numbers is a function whose domain is the set of natural numbers 1, 2,…, k,… and whose range is contained in . Thus, a sequence of real numbers can be viewed as a set of numbers {x1, x2,…, xk,…}, which is often also denoted as {xk} (or sometimes as , to indicate explicitly the range of values that k can take).

A sequence {xk} is increasing if x1 < x2 < · · · < xk · · ·; that is, xk < xk+1 for all k. If xkxk+1, then we say that the sequence is nondecreasing. Similarly, we can define decreasing and nonincreasing sequences. Nonincreasing or nondecreasing sequences are called monotone sequences.

A number x* is called the limit of the sequence {xk} if for any positive ε there is a number K (which may depend on ε) such that for all k > K, |xkx*| < ε; that is, xk lies between x* − ε and x* + ε for all k > K. In this case we write

equation

or

A sequence that has a limit is called a convergent sequence.

The notion of a sequence ...

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