488 sums of powers

Computing sums of powers

2

1

––

1 – x

are often interested in problems involving the sum of a

large collection of quantities, and a compact notation for

writing such sums is helpful. The Greek letter Σ(repre-

senting “S” for “sum”), the so-called sigma notation

invented by Swiss mathematician L

EONHARD

E

ULER

in

1755, can be used to express the

sum of the numbers a1,a2,…,anas . One reads this

as “the sum from k= 1 to nof ak,” and computes it

via the following procedure:

Write the terms ak, first with kreplaced by 1,

and then with kreplaced by 2, stopping with

kreplaced by n. Add together all the terms

written: = a1+ a2+ … + an

Thus, for example, the expression represents the

sum 12+ 22+ 32+ 42, which equals 30.

The letter kused as a subscript in sigma notation is

called the index of summation and is a

DUMMY VARI

-

ABLE

: any other letter could be used in its stead. For

example, the expressions and also represent

the same sum 12+ 22+ 32+ 42.

A summation need not begin with index k= 1. For

example, the expression represents the sum

(103– 102) + (113– 112) + (123– 122). A summation

could contain just a single term, as for , for

instance. Any summation of the form with n< m

is considered empty and to have value zero. Finally,

sums alternating in sign can be represented by intro-

ducing a factor of the form (–1)k–1. For example,

equals .

Summations satisfy the following two basic prop-

erties:

where cis a constant. (These statements are patently

true when the summations are written out in full.)

The sigma notation is also used to denote

SERIES

,

that is, infinite sums:

For example, the

GEOMETRIC SERIES

can be written:

. If –1 < x< 1, then it has

value , in which case we can write: .

G

OTTFRIED

W

ILHELM

L

EIBNIZ

(1646–1716), coin-

ventor of

CALCULUS

, used an elongated “S,” ∫, both for

the sum of a sequence of integers and as a sign of inte-

gration (which he thought of as continuous summation).

See also

ARITHMETIC SERIES

;

CONVERGENT SERIES

;

INFINITE PRODUCT

;

SUMS OF POWERS

.

sums of powers The following formulae, called the

sums of powers formulae, give simple equations for the

sums of the first few counting, square, and cube num-

bers. They can readily be proved by the method of

INDUCTION

:

It is also possible to establish these formulae geomet-

rically.

The validity of the first formula can be seen by

dividing a n×(n+ 1) grid of squares into two “stair-

cases,” each representing the same sum 1 + 2 + 3 +…+ n.

This gives: