Concrete mathematics : a foundation for computer science

22 SUMS
The three-dots notation has many uses, but it can be ambiguous and a “Le signe ,T~~~
bit long-winded. Other alternatives are available, notably the delimited form indique Ve
/‘on doit dormer
au nombre entier i
(2.2) to&es ses valeurs
k=l
1,2,3 ,..., et
prendre la somme
which is called Sigma-notation because it uses the Greek letter t (upper- des termes.”
case sigma). This notation tells us to include in the sum precisely those
terms ok whose index k is an integer that lies between the lower and upper
limits 1 and n, inclusive. In words, we “sum over k, from 1 to n.” Joseph
Fourier introduced this delimited t-notation in 1820, and it soon took the
mathematical world by storm.
Incidentally, the quantity after x (here ok) is called the summa&.
The index variable k is said to be bound to the x sign in (2.2), because
the k in ok is unrelated to appearances of k outside the Sigma-notation. Any
- J. Fourier I1021
other letter could be substituted for k here without changing the meaning of Well, I wouldn’t
(2.2). The letter i is often used (perhaps because it stands for “index”), but
we’ll generally sum on k since it’s wise to keep i for &i.
It turns out that a generalized Sigma-notation is even more useful than
want to use a Or n
as the index variable
instead of k in
(2.2); those letters
the delimited form: We simply write one or more conditions under the x., are “free variables”
to specify the set of indices over which summation should take place. For
example, the sums in (2.1) and (2.2) can also be written as
that do have mean-
mg outside the 2
here.
ix ak . (2.3)
l