Abstract
Infinite Series
The goal of the talk 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 17th 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. This problem is highlighted in
the first volume of Polya's
"Mathematics and Plausible Reasoning,
Volume 1" in his discussion on the use
of analogy leading to discovery. It is
this method of `solving' the problem
that this talk will address. Along the
way we will discuss many of the items
touched upon at a more advanced level
last week in Andrew's talk on
generating functions; geometric series,
telescoping series, and Maclaurin
series.