An Illustration of the Sum of an Infinite Series.
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.
The 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, 34210DavutpasaIstanbul,
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.


Hallmark Series

Who did it?



Limit and
Convergence Graphs

1.


Oresme
(13231349)




2.


Taylor (16851731)



3.


Leibniz
(16461716)




4.


Euler (17071783)




Useful Links and Books

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: < http://curvebank.calstatela.edu/hseries/hseries.htm
> (Streaming video).

For the Geometric
Series:
< http://curvebank.calstatela.edu/series/series.htm
> (Streaming video). 
For Baravelle Spirals: < http://curvebank.calstatela.edu/baravelle/baravelle.htm
> (Java animation).

For another Geometric Series
< http://curvebank.calstatela.edu/seriesfold/seriesfold.htm
> (Paper folding demonstration).

McQuarrie, Donald A., Mathematical Methods
for Scientists and Engineers,
University Science Books, 2003, pp. 63113.

Stewart, James, Calculus, 5th ed., Thomson:
Brooks/Cole, 2003, pp. 736827.




MATHEMATICA^{®}
animation contributed by
Alexander Gomez
2009.


