Sequences and Series— in which we discover very odd behaviour in even the smallest infinite set
Much of the effort we have expended in developing the system ℝ of real numbers and in describing the infinite can now be put to good use. In this chapter we shall be investigating limiting processes, the very foundation of analysis. We begin by agreeing that an infinite sequence is a countably infinite set of real numbers occurring in some definite order, a1,a2,a3, …, a n , …. Each a i ∈ℝ and there is one a i for each i∈ℕ. A favoured abbreviation for a sequence is (a n ), where a n denotes the nth term of the sequence.
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Suggestions for Further Reading
- J. A. Green, Sequences and Series, Routledge and Kegan Paul (1966). Packed full of examples of sequences and series.Google Scholar
- H. E. Huntley, The Divine Proportion, Dover (1970). If your appetite for the golden ratio has been whetted by this chapter, Huntley’s book goes a long way towards satisfying it.Google Scholar