lec12 - Math 117 12 Convergence of sequences A sequence is...

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Math 117 – May 12, 2011 12 Convergence of sequences A sequence is a function whose domain is the set of natural numbers N . We usually denote the values of the function by s n , instead of s ( n ), and call these values the terms of the sequence . We will be primarily interested in sequences of real numbers, in which case the terms of the sequence s n R for every n N . We use the notation ( s n ) for the entire sequence. Definition 12.1. A sequence ( s n ) is said to converge to s R , if for every ± > 0 there exists n N , such that for every n > N , | s n - s | < ± . The number s is called the limit of the sequence, and is denoted by s = lim s n . If the limit exists, ( s n ) is called a convergent sequence. If a sequence does not converge to any real number, it is said to be divergent. When trying to show that the real number s is the limit of the sequence ( s n ), it is sometimes more convenient to consider the sequence made of the differences ( s n - s ). The following theorem illustrates
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