2.2 Some Summation Formulas

Figure (a) shows two polygons, each having area . If we slide
the
two polygons so that they touch, we create a rectangle as in figure (b) whose
area is . Thus

i.e.,

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

Hence

Add the left sides of these equations together, and equate the result to the sum of the right sides:

In the left side of this equation all of the terms except the first cancel. Thus

so

and

This completes the second proof of (2.8).

To find
we use the same sort of argument. For any real
number we have

Hence,

Next we equate the sum of the left sides of these equations with the sum of the right sides. As before, most of the terms on the left side cancel and we obtain

We now use the known formula for :

so

and finally

You should check that this formula agrees with the calculations made in (2.7). The argument we just gave can be used to find formulas for , and for sums of higher powers, but it takes a certain amount of stamina to carry out the details. To find , you could begin with

Add together the results of this equation for and get

Then use equations (2.8) and (2.9) to eliminate and , and solve for .

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]