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Another Converging Series

" Read Euler. He is our master in all. "
Laplace ( 1749 - 1827 )
Euler stamp

An Illustration of the Sum of an Infinite Series.
Convergent alternating series

Large square divided into 9 equal squares
minus one-third
. . . plus one-ninth
More of the equation
More of the equation More of the equation More of the equation Conclusion

This MATHEMATICA® project is a step by step visual demonstration that the sum of this alternating infinite series converges.  Each square graphic is labeled on top with a value that depicts the summation at a certain stage.  There are a total of eight square graphics in this illustration.

MATHEMATICA® images were created by Alexander Gomez, a student at CSU Los Angeles.  The initial idea was based on "Proof Without Words" in "The College Mathematics Journal", 40 (1), January, 2009, p. 39 by Hasan Unal, Yildiz Technical University, Dept. of Mathematics, 34210-Davutpasa-Istanbul, Turkey.     The NCB thanks both Alexander and Hasan for their work.

        Historical Sketch:

The most basic of sequences and infinite series has its roots in Antiquity.  From Egypt we find the sum of a finite power series is in the Rhind Papyrus, Problem 79.  Zeno's Paradox of Achilles and the Tortoise has had a life span of over 2,000 years.  One of the few high points for mathematics in the Middle Ages was Oresme's grouping of fractions to show the harmonic series would never converge. 

With accumulated perspective, one of the legacies of centuries of thought is far more patience and curiosity about infinite processes.  Certainly this  gave birth  to the "calculus"
in the seventeenth century.  Today we apply the most rigorous ideas of limit and convergence.  These concepts were not developed until the late eighteenth and early nineteenth centuries.  The pure usefulness of a series that converges remains important to all theoreticians.  And today, well, we can add the embellishment of animation and computer graphics.

Animated gif file

Hallmark Series
Who did it?
Limit and Convergence Graphs
Oresme  (1323-1349)

Taylor series Taylor (1685-1731)

Leibniz Leibniz (1646-1716)

Convergence image
Euler Euler (1707-1783)

Useful Links and Books
School of Athens stamp

One of the pleasures of visiting the Sistine Chapel in Vatican City is seeing Raphael's The School of Athens.  His famous frescoes are just outside the doorway to the Chapel.  Among the important mathematical figures represented are Euclid, Ptolemy, Pythagoras and Zeno.  Click on the above stamp to see an enlargement. 

In particular, The School of Athens is considered one of the earliest and finest examples of perspective, a highly geometrical illusion of giving distance its proper proportion on a plane surface.

Hasan Unal,  Proof Without Words:Sum of an Infinite Series, The College Mathematics Journal, 40 (1), January, 2009, p. 39.

For the Harmonic Series: < > (Streaming video).

For the Geometric Series:  >  (Streaming video).
For Baravelle Spirals:  < > (Java animation).
For another Geometric Series  < >  (Paper folding demonstration).
McQuarrie, Donald A.,  Mathematical Methods for Scientists and Engineers, University Science Books, 2003, pp. 63-113.
Stewart, James, Calculus, 5th ed., Thomson: Brooks/Cole, 2003, pp. 736-827.
Leibniz Nexton Bessel
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MATHEMATICA®  animation contributed by

Alexander Gomez