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Understanding Infinite Series in the Calculus

  Professor Michael Krebs
Jeffrey L. Derbidge of CSULA
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Harmonic Series

Harmonic series equation

Photo: Krebs and Derbidge
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Click on each image to reveiw the steps for determing the Harmonic Series is divergent
Equations #1
Equations #2
Equations #3
Equations #4

In the harmonic series, the sum of each group of fractions being added is greater than one-half. Thus, the harmonic series is constantly adding terms greater than any assignable limit.  We say the harmonic series diverges.

Convergence occurs only if the various sums "taken, from the first, in whatever number one wishes, finish by constantly have an absolute value less than any assignable limit."

Cours d'analyse de l'Ecole Royale Polytechnique, VI,  Augustin-Louis Cauchy

[From Cauchy's lecture notes.]
But as always in mathematics, be careful.  The alternating harmonic series converges.
Alternating harmonic series converges.
This is a result of the Taylor series of the natural logarithm.                    

Series Quiz:  Identify the famous investigator.

1.    Quiz #1   .

    Quiz #2   .

3.    Quiz #3   .

  Quiz #4   .

5.    Quiz #5   .

6.    Quiz #6   .

Useful Links and Books
James Stewart, Calculus, 5th ed,. Thomson Brooks/Cole, 2003, p. 753-754.
Howard Eves, An Introduction to the History of Mathematics, 6th ed,. The Saunders College Publishing, 1990, p. 264.
Victor J. Katz, A Brief History of Mathematics, BRIEF VERSION, Pearson Addison Wesley, 2004, pp. 208-210.
Eric W. Weisstein, "Series" in CRC Concise Encyclopedia of MATHEMATICS, CRC Press, 1998,  p. 800 and p. 1618.  
We also recommend reading the article "Book Stacking Problem." p. 155. 
Professors have illustrated the harmonic series using stacks of books.
Stack of blocks illustration
Harmonic Series
     James Lesko, Sums of Harmonic-Type Series, The College Mathematics Journal, Vol.35, No.3, May 2004, pp. 171-182.
     Curtis Feist and Ramin Naimi, Almost Alternating Harmonic Series, The College Mathematics Journal, Vol.35, No.3, May 2004, pp. 183-191.
Full Proofs
     K. Knopp, Theory and Application of Infinite Series, Hafner, 1951. Link to another web site

Both the Harmonic Series and the harmonica are named for a property of sound studied by the ancient Pythagoreans.   Given a vibrating string of length one unit, a string of 1/2 the length will vibrate with a frequency of twice the basic unit.  Similarly a length of 1/3 will vibrate with a frequency of three times the fundamental unit and so on.  The combination of frequencies produces musical harmony; thus the name.
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William Stellmacher
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Wing T. (Sterling) Wong

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