Transcription of Sequences and Series - Michigan State University
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Sequences and Series - Michigan State University
Chapter 2 Sequences and The Limit of a SequenceDefinition a function whose domain this function is denoted byf, then the valuesf(n) (n N) determine the sequence uniquely, and vise-versa. Therefore,a sequence is usually denoted by(a1,a2,a3,a4, ) or (an) n=1,wherean=f(n) forn this course we only study Sequences of real numbers; namely functionsf:N of the following are common ways to describe a sequence .(i) (1,12,13,14, ),(ii) (nn+1) n=1,(iii) (an), wherean= 2nfor alln N,(iv) (xn), wherex1= 2 andxn+1=xn+12. This is the induction way or recursion wayto define a the difference between a sequence (an) and a set{an:n N}:(( 1)n) n=1= ( 1,1, 1,1, 1,1, ) is a sequence , having infinitely many terms (whichcan have repeated values);{( 1)n:n N}={1, 1}is simply a set of two elements, not a countable set nor asequence;(c) = (c,c,c,c, ) is the constant sequence ;{c}is the set of single (Convergence of a sequence ).
2 2. Sequences and Series A topological way to say lima n = ais the following: Given any -neighborhood V (a) of a, there exists a place in the sequence after which all of the terms are in V (a): Easy Fact: lim(c) = cfor all constant sequences (c): Quanti ers. The de nition of lima n = aquanti es the closeness of a n to aby an arbi-
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