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Series - math.ucdavis.edu

Chapter 4 SeriesDivergent Series are the devil, and it is a shame to base on themany demonstration whatsoever. (Niels Henrik Abel, 1826)This Series is divergent, therefore we may be able to do somethingwith it. (Oliver Heaviside, quoted by Kline)In this chapter, we apply our results for sequences to Series , or infinite convergence and sum of an infinite Series is defined in terms of its sequence offinite partial convergence of seriesA finite sum of real numbers is well-defined by the algebraic properties ofR, but inorder to make sense of an infinite Series , we need to consider its convergence .

A condition for the convergence of series with positive terms follows immedi-ately from the condition for the convergence of monotone sequences. Proposition 4.6. A series P a nwith positive terms a n 0 converges if and only if its partial sums Xn k=1 a k M are bounded from above, otherwise it diverges to 1. Proof. The partial sums S n= P n k=1 a

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