Figure (a) shows two polygons, each having area . If we slide
two polygons so that they touch, we create a rectangle as in figure (b) whose
area is . Thus
The proof just given is quite attractive, and a proof similar to this was probably known to the Pythagoreans in the 6th or 5th centuries B.C. Cf [29, page 30]. The formula itself was known to the Babylonians much earlier than this[45, page 77], but we have no idea how they discovered it.
The idea here is special, and does not generalize to give a formula for . (A nice geometrical proof of the formula for the sum of the first squares can be found in Proofs Without Words by Roger Nelsen[37, page 77], but it is different enough from the one just given that I would not call it a ``generalization''.) We will now give a second proof of (2.8) that generalizes to give formulas for for positive integers . The idea we use was introduced by Blaise Pascal [6, page 197] circa 1654.
For any real number , we have
we use the same sort of argument. For any real
number we have
Jacob Bernoulli (1654-1705) considered the general formula for power sums. By using a technique similar to, but slightly different from Pascal's, he constructed the table below. Here is denoted by , and denotes a missing term: Thus the in the fourth line of the table below indicates that there is no term, i.e. the coefficient of is zero.
Thus we can step by step reach higher and higher powers and with slight effort form the following table.
Whoever will examine the series as to their regularity may be able to continue the table[9, pages 317-320]. 2.1
He then states a rule for continuing the table. The rule is not quite an explicit formula, rather it tells how to compute the next line easily when the previous lines are known.
A formula for was proved by Archimedes (287-212 B.C.). (See Archimedes On Conoids and Spheroids in [2, pages 107-109]). The formula was known to the Babylonians[45, page 77] much earlier than this in the form
A technique for calculating general power sums has been known since circa 1000 A.D. At about this time Ibn-al-Haitham, gave a method based on the picture below, and used it to calculate the power sums up to . The method is discussed in [6, pages 66-69]