{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

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

Info icon This preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon
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 /x dx . 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
Image of page 1

Info icon This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
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 . Since the subsequence { H 2 k } is unbounded, the sequence { H n } diverges. squaresolid Using the same type of argument, one can show that for any positive integer M , H M k 1 + k parenleftbigg M - 1 M parenrightbigg .
Image of page 2
Image of page 3
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern