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Unformatted text preview: The Harmonic Series Diverges Again and Again * Steven J. Kifowit Prairie State College Terra A. Stamps Prairie State College The harmonic series, ∞ summationdisplay n =1 1 n = 1 + 1 2 + 1 3 + 1 4 + 1 5 + ··· , is one of the most celebrated infinite series of mathematics. As a counterexam ple, few series more clearly illustrate that the convergence of terms to zero is not sufficient to guarantee the convergence of a series. As a known series, only a handful are used as often in comparisons. From a pedagogical point of view, the harmonic series provides the instructor with a wealth of opportunities. The leaning tower of lire (Johnson 1955) (a.k.a the book stacking problem) is an interesting handson activity that is sure to surprise students. Applications such as Gabriel’s wedding cake (Fleron 1999) and Euler’s proof of the divergence of ∑ 1 /p ( p prime) (Dunham 1999, pages 70–74) can lead to some very nice discussions. And the proofs of divergence are as entertaining as they are educational. A quick survey of modern calculus textbooks reveals that there are two very popular proofs of the divergence of the harmonic series: those fashioned after the early proof of Nicole Oresme and those comparing ∑ n k =1 1 /k and integraltext n +1 1 1 /xdx . While these proofs are notable for their cleverness and simplicity, there are a number of other proofs that are equally simple and insightful. In this article, the authors survey some of these divergence proofs. Throughout, H n is used to denote the n th partial sum of the harmonic series. That is, H n = 1 + 1 2 + 1 3 + ··· + 1 n , n = 1 , 2 , 3 , . . . . A common thread connecting the proofs is their accessibility to firstyear calcu lus students. * Appears in The AMATYC Review , Vol. 27, No. 2, Spring 2006, pp. 31–43 1 The Proofs Though the proofs are presented in no particular order, it seems fitting to begin with the classical proof of Oresme. Proof 1 Nicole Oresme’s proof dates back to about 1350. While the proof seems to have disappeared until after the Middle Ages, it has certainly made up for lost time. Proof: Consider the subsequence { H 2 k } ∞ k =0 . H 1 = 1 = 1 + 0 parenleftbigg 1 2 parenrightbigg , H 2 = 1 + 1 2 = 1 + 1 parenleftbigg 1 2 parenrightbigg , H 4 = 1 + 1 2 + parenleftbigg 1 3 + 1 4 parenrightbigg > 1 + 1 2 + parenleftbigg 1 4 + 1 4 parenrightbigg = 1 + 2 parenleftbigg 1 2 parenrightbigg , H 8 = 1 + 1 2 + parenleftbigg 1 3 + 1 4 parenrightbigg + parenleftbigg 1 5 + 1 6 + 1 7 + 1 8 parenrightbigg > 1 + 1 2 + parenleftbigg 1 4 + 1 4 parenrightbigg + parenleftbigg 1 8 + 1 8 + 1 8 + 1 8 parenrightbigg = 1 + 3 parenleftbigg 1 2 parenrightbigg . In general, H 2 k ≥ 1 + k parenleftbigg 1 2 parenrightbigg ....
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This note was uploaded on 01/04/2011 for the course MATH 128A taught by Professor To during the Spring '10 term at SUNY Geneseo.
 Spring '10
 To
 Harmonic Series

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