Concrete mathematics : a foundation for computer science

Sure: 1 + 2 +
4 + 8 + . . is the
“infinite precision”
representation of
the number -1,
in a binary computer
with infinite
word size.
2.7 INFINITE SUMS 57
the fact that sums can be infinite. And the truth is that infinite sums are
bearers of both good news and bad news.
First, the bad news: It turns out that the methods we’ve used for manipulating
1’s are not always valid when infinite sums are involved. But next,
the good news: There is a large, easily understood class of infinite sums for
which all the operations we’ve been performing are perfectly legitimate. The
reasons underlying both these news items will be clear after we have looked
more closely at the underlying meaning of summation.
Everybody knows what a finite sum is: We add up a bunch of terms, one
by one, until they’ve all been added. But an infinite sum needs to be defined
more carefully, lest we get into paradoxical situations.
For example, it seems natural to define things so that the infinite sum
s = l+;+;+f+&+&+...
is equal to 2, because if we double it we get
2s = 2+1+;+$+;+$+.- = 2+s.
On the other hand, this same reasoning suggests that we ought to define
T = 1+2+4+8+16+32-t...
to be -1, for if we double it we get
2T = 2+4+8+16+32+64+... = T-l.
Something funny is going on; how can we get a negative number by summing
positive quantities? It seems better to leave T undefined; or perhaps we should
say that T = 00, since the terms being added in T become larger than any
fixed, finite number. (Notice that cc is another “solution” to the equation
2T = T - 1; it also “solves” the equation 2S = 2 + S.)
Let’s try to formulate a good definition for the value of a general sum
x kEK ok, where K might be infinite. For starters, let’s assume that all the
terms ok are nonnegative. Then a suitable definition is not hard to find: If
there’s a bounding constant A such that
for all finite subsets F c K, then we define tkeK ok to be the least such A.
(It follows from well-known properties of the real numbers that the set of
all such A always contains a smallest element.) But if there’s no bounding
constant A, we say that ,YkEK ok = 00; this means that if A is any real
number, there’s a set of finitely many terms ok whose sum exceeds A.