Abstract
If a sequence of numbers is approximating more closely to some fixed real number, this number is called “the limit of the sequence.” This chapter examines what is meant by a limit and the conditions under which sequences have limits; it derives the limits of many important sequences. A sequence is a list of real numbers arranged like the list of natural numbers. The numbers that make up the sequence might be different, but need not be. A sequence is a function whose domain is the set of natural numbers and whose range is some subset of the real numbers. Sequences arise as sets of closer and closer approximations to some real number. Sequences are constructed from other sequences, such as, by multiplying the corresponding terms of two sequences. The chapter examines the relationship between the limits of sequences, which are related in this and other ways. Monotonic sequences are of great importance because it is easy to test whether they are convergent.
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