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
12+22+32+...+n2. 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:
Faulhaber'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 ∑n25 although he published the results only up to ∑n17.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 ∑n23 were reliable." The computations Faulhaber would have done, by hand, are difficult and involve rather large integers. Any participant in today's class can use the techniques introduced today together with Maple to verify a sum formula for 25th powers and this formula would have integers like 3,606,964,705 and 1,181,820,455 (really!).
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 17th 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.