sums of powers 489

2×(1 + 2 + 3 +…+ n) = n×(n+ 1)

In a similar vein, one can show that six copies of a

three-dimensional “staircase” stack together to form an

n×(n+ 1) ×(2n+ 1) rectangular box, thereby estab-

lishing the validity of the sum-of-squares formula.

Finally, each L-shaped region in a subdivided (1 + 2

+…+ n) ×(1 + 2 +…+ n) square has area given by a

cube number. Summing together these areas establishes

the sum-of-cubes formula.

A

RCHIMEDES OF

S

YRACUSE

(ca. 287–212

B

.

C

.

E

.)

knew the formula for the sum of the first ncounting

numbers. Arab scholars, who translated and pre-

served the works of Archimedes, were thus also

aware of the result. C

ARL

F

RIEDRICH

G

AUSS

(1777–1855) is said to have discovered the same for-

mula as a school student, employing a clever trick of

writing the same sum forward and backward. (This is

now considered a standard technique in the study of

ARITHMETIC SEQUENCES

.)

In 1713 Swiss mathematician Jacques Bernoulli of

the famous B

ERNOULLI FAMILY

searched for a general

formula for the sum of the first few kth powers: 1k+ 2k

+ 3k+…+ nk. He noticed that this sum appears as the

coefficient of xkin the T

AYLOR SERIES

of

k!(1 + ex+ e2x+…+ enx)

(To see this, write out each term emx as a series 1 + mx

+ +… Then add and collect all terms containing

the expression xk.) Using the formula for the

sum of a

GEOMETRIC SEQUENCE

, this expression can

be rewritten as:

The first factor here is k!, and the Taylor series of the

second factor is straightforward:

Unfortunately, the Taylor series of the third factor is

difficult to compute. Bernoulli decided to simply write:

for some numbers B0, B1, B2, … yet to be determined.

Given this, it now follows that 1k+ 2k+ 3k+…+ nkis

the coefficient of xkin the product:

Expanding brackets and identifying the coefficient of

xkin this product eventually leads to the remarkable

formula:

where a quantity Bris to be interpreted as the value Br.

The numbers B0, B1, B2, … are today known as the

Bernoulli numbers, and their values are given by:

with Bk= 0 for all odd values of kgreater than 1.

Thus, as an example, we again have:

The Bernoulli numbers have since been studied

extensively, and many of their properties are now well

understood. These numbers make an appearance in

evaluating the

ZETA FUNCTION

at even values. Specifi-

cally, mathematicians have shown that: