9.4 POWER SERIES AND EULERIAN MATHEMATICS 347- The early stages of any investigation,
- The early stages of any person's mathematical education.
We certainly don't mean that rigor is evil, but we do wish to stress that lack of rigor
is not the same as nonsense. A fuzzy, yet inspired idea may eventually produce a
rigorous proof; and sometimes a rigorous proof completely obscures the essence of an
argument.
There is, of course, a fine line between a brilliant, non-rigorous argument and
poorly thought-out silliness. To make our point, we will give a few examples of "Eu
lerian mathematics," which we define as non-rigorous reasoning that may even be (in
some sense) incorrect, yet leads to an interesting mathematical truth. We name it in
honor of the 18 th-century Swiss mathematician Leonhard Euler, who was a pioneer
of graph theory and generatingfunctionology, among other things. Euler's arguments
were not always rigorous or correct by modem standards, but many of his ideas were
incredibly fertile and illuminating.
Most of Euler's "Eulerian" proofs are notable for their clever algebraic manipula
tions, but that is not the case for all of the examples below. Sometimes a very simple
yet "wrong" idea can help solve a problem.1 3 We will begin with two examples (by
students at the University of San Francisco) that help to solve earlier problems. They
are excellent illustrations of the "bend the rules" strategy discussed on page 20.
Example 9.4.5 Solve Problem 1.3.8 on page 9:For any sequence of real numbers A = (a I , a 2 , a 3 ,.
..), define ..1A to be
the sequence (a 2 -al ,a 3 - a 2 ,a 4 - a 3 , ... ) whose nth term is an+ 1 -an.
Suppose that all of the terms of the sequence ..1 (..1A) are 1, and that
al 9 = a 94 = O
.Find al.
Partial Solution: Even though this is not a calculus problem-the variables are
discrete, so notions of limit make no sense-we can apply calculus-style ideas. Think
of A as a function of the subscript n. The..1 operation is reminiscent of differentiation;
thus the equation
..1 (..1A) = (1, 1, 1, ... )suggests the differential equation
d^2 A
dn^2
= (^1).
Solving this (pretending that it makes sense) yields a quadratic function for n. None
of this was "correct," yet it inspires us to try guessing that an is a quadratic function of
n. And this guess turns out to be correct!
Example 9.4.6 Solve Problem 1.3. 12 on page 9:
(^13) The 20th-century king of algebraic Eulerian thinking was the self-educated Indian mathematician S. Ra
manujan, who worked without any rigor, yet made many incredible discoveries in number theory and analysis.
See [38] for details. Recently, mathematics has begun to see a movement away from rigor and toward intuition
and visualization; perhaps the most eloquent statement of this approach is the preface to [29].