Although the rigor of this talk will
not be stringent enough for
mathematicians, the material is
quite interesting and the techniques
quite useful. I do not know infinite
series well enough to
say that I truly understand the topic,
but I enjoy investigating it in the
dozens of books
on infinite series in my library and
revelling in the amazing results.
This ambivalent feeling about infinite
series probably stems from the fact
that I did not understand
anything about infinite series in
my calculus class 37 years ago and
the other fact that
I must now teach infinite series every
year to my students. One of the
highlights of the year (at least
for me) in my class, after the Advanced
Placement exam in May, is to see
how the
28 year old Euler in 1735 solved a
problem known as the Basel Problem. This
problem had been
proposed by Jakob Bernoulli in 1689
when he collected
the all of the work on infinite series
of the 
century in a volume
entitled
Tractatus De Seriebus Infinitis.
Bernoulli's comment in this volume
was that the
evaluation ``is more difficult than one
would expect''. Little did he know
how difficult it really was and that
it would take almost 50 years for the
problem to be solved. In my class, we
then see how the problem can be solved
to the satisfaction of mathematicians
today, using only knowledge gained in
first year calculus.
This problem is highlighted in the
first volume of Polya's
Mathematics and Plausible
Reasoning, Volume 1 [6] in his
discussion on the use of analogy
leading to discovery.
It is this method of `solving' the
problem that this talk will address. The
real proofs using
elementary calculus are many, but not
appropriate for this circle. If you
have studied calculus
then you will find proofs and
references to proofs in [2],
[9], and an extensive
bibliography is given by E. L. Stark in
Mathematics Magazine,
47 (1974) pp 197-202.
A word of caution is in order about
entering fields without a proper
foundation. It is easy to
do things by example and never attain an
understanding. This is the issue of
How versus
Why that Paul Zeitz gave a talk
on last year. It is interesting to
see the remarks of
G. Chrystal in Textbook of
Algebra [3] written in 1889. ``A
practice has sprung
up of late (encouraged by demands for
premature knowledge in certain
examinations) of hurrying
young students into the manipulation of
the machinery of the Differential
and Integral Calculus
before they have grasped the preliminary
notions of a Limit and of an
Infinite
Series, on which all the meaning and
all the uses of the Infinitesimal
Calculus are based. Besides
being a sham, this is a sin against the
spirit of mathematical progress.''
It is remarkable to
see over one hundred years later that we
are still fighting this battle. On
the other hand, it is never too soon to
begin thinking about big ideas and
seeing how the
great minds approached them. In this we
will follow the great mathematician
Pierre-Simon de Laplace:
``Read Euler, read Euler. He is the master of us all.''