harmapa - The Harmonic Series Diverges Again and Again *...

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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 hands-on 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 first-year 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.

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harmapa - The Harmonic Series Diverges Again and Again *...

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