I'll describe one traditional way, very old, to write a formula for
1+2+3+...+n when n is a positive integer. And then, in front of the
astonished (well, maybe) class, I will attempt to show how such a
formula could be discovered and *proved* using a tool like `Maple`. The discovery part is interesting and
fun, but the word *proved* is one which takes some understanding
and belief, and, to me, proving such a result with a machine is not at
all obvious. I do not mean computing the sum of the first 276
integers (that's 38,226) or the first 10,017 integers (that's
50,175,153) but an effort to get an algebraic method which shortcuts
the whole summation and allows the sum to be computed briefly and
efficiently. And this algebraic method will be valid for *any
positive integer n*. So we will need to use `Maple` carefully. I should then hand out a discussion of the some of the
commands used, maybe without the errors I'll probably commit in
class.

**Student challenge; student work**

I'd like students to "discover" in a similar way a formula for the sum
of the first n squares. That is, find some simpler (to write, and to
compute!) expression for
1^{2}+2^{2}+3^{2}+...+n^{2}. I'll try
to walk around the room and "facilitate" your explorations. I will
strongly urge you to work together and to discuss what you are
doing. In fact, it may be useful for students to work in pairs.

Depending on the pace of the class, there may be time for a student
(or a pair of students) to show the whole class a discovered
solution. I think that would be *neat*.

**Further historical background**

Formulas for the sums of integers, and squares of integers, etc. have
been known for a long time. Besides their obvious interest as sheer
curiosities, these formulas can be used to compute certain quantities
with the ideas of calculus. Historically, the first systematic listing
of these formulas seems to be due to Johann Faulhaber about three and
a half centuries ago. Faulhaber found formulas for the sums of powers
of integers when the power ranged from 1 to 25. Remember, this is all
a long time ago, and he must have done it by hand. A biography of Johann
Faulhaber, 1580-1635 declares

He gives the formulae in the form of a secret code, which was common practice at the time. ... Faulhaber had the correct formulae up to k=23, but his formulae for k=24 and k=25 appear to be wrong.The person who verified this statement, and who worked to "decode" Faulhaber's concealed formulas, was Donald Knuth. It is nice to refer to a living person, and Donald Knuth, 1938-- is probably the world's most eminent computer scientist. His original training was in mathematics, but, perhaps in spite of that, his writing is very informative and can be quite entertaining. Much of Knuth's 1993 paper, Johann Faulhaber and sums of powers, could probably be understood by most of the students in this class. It has the following paragraph:

In much scientific and technical work, the ∑ symbol is used as an abbreviation for summation, that is, for adding up things. Knuth discusses both the formulas and the encryption scheme and later concludes (rather sadly, I think), "Therefore we have no evidence that Faulhaber's calculations beyond ∑nFaulhaber's cryptomath.Mathematicians of Faulhaber's day tended to conceal their methods and hide results in secret code. Faulhaber ends his book [2] with a curious exercise of this kind, evidently intended to prove to posterity that he had in fact computed the formulas for sums of powers as far as ∑n^{25}although he published the results only up to ∑n^{17}.

**More history, including names of formulas**

Probably the type of formulas discussed here should use Faulhaber's
name, but they don't. They frequently are called *Bernoulli*
formulas after some member(s) of a 17^{th} century Swiss
family of mathematicians, probably Jacob
Bernoulli, 1654-1705. The formulas can be written in terms of what
are called the Bernoulli numbers. Much more than almost anyone would
want to know about these numbers is here, with a briefer
Wikipedia article here.

The Wikipedia article contains an assertion which I've also seen
elsewhere and is a bit remarkable. The case can be made that the first
computer program was written to compute Bernoulli numbers, and
therefore to get the summation formulas discussed in this class. Augusta
Ada King, Countess of Lovelace (!), 1815-1852, was the daughter of
Lord Byron, an English poet. See here for
another biography. She became acquainted with Charles Babbage, 1791-1871. Babbage invented and
supervised the construction of the *Difference
Engine*, an early digital computational device, and planned an
*Analytical
Engine* which had many logical features in common with the
stored program digital computers first built 75 years later. The
latter's design was partially based on complicated Jacquard looming
machines! Countess Lovelace wrote essays about these machines and
described how to compute Bernoulli numbers with such a machine, a
rather remarkable achievement for that time. She died of cancer at age
37. I've always thought she was an inspiration for the marvelously
talented young English woman mathematician in Tom Stoppard's wonderful
play, *Arcadia*.

**The next homework assignment**

This handout has some advice
about how to discover and verify some summation formulas more briefly
than what we've already done. There are no great improvements, but
rather a desire to use some of `Maple`'s
facilities more so that there is less need for human intervention (and
so less likelihood for human error!).

Read about what I'd like you to do. There are seven homework
questions and seven **solution teams**. Please find the other
people on your team and solve your question together and get the
answer to me next time. And maybe send e-mail telling me if you want
to present your solution.

**
Maintained by
greenfie@math.rutgers.edu and last modified 9/15/2008.
**