Concrete mathematics : a foundation for computer science

58 SUMS
The definition in the previous paragraph has been formulated carefully
so that it doesn’t depend on any order that might exist in the index set K.
Therefore the arguments we are about to make will apply to multiple sums
with many indices kl , k2, . . , not just to sums over the set of integers.
In the special case that K is the set of nonnegative integers, our definition
for nonnegative terms ok implies that
Here’s why: Any nondecreasing sequence of real numbers has a limit (possibly
ok). If the limit is A, and if F is any finite set of nonnegative integers
whose elements are all 6 n, we have tkEF ok 6 ~~Zo ok < A; hence A = co
or A is a bounding constant. And if A’ is any number less than the stated
limit A, then there’s an n such that ~~=, ok > A’; hence the finite set
F ={O,l,... ,n} witnesses to the fact that A’ is not a bounding constant.
We can now easily com,pute the value of certain infinite sums, according
to the definition just given. For example, if ok = xk, we have
In particular, the infinite sums S and T considered a minute ago have the respective
values 2 and co, just as we suspected. Another interesting example is
k5 n
= l.im~k~=J~m~_l = l.
n-+cc
k=O
0
Now let’s consider the ‘case that the sum might have negative terms as
well as nonnegative ones. What, for example, should be the value of
E(-1)k = l-l+l--l+l-l+~~~?
The set K might
even be uncountable.
But only a
countable number
of terms can
be nonzero, if a
bounding constant
A exists, because at
most nA terms are
3 l/n.
k>O
“Aggregatum
If we group the terms in pairs, we get
quantitatum
a-a+a-a+a--a
etc. nunc est = a,
(l--1)+(1-1)+(1-1)+... = O+O+O+... ) nunc = 0, adeoque
continuata in infiniso
the sum comes out zero; but if we start the pairing one step later, we get turn serie ponendus
‘-(‘-‘)-(1-1)-(1-l)-... = ‘ - O - O - O - . . . ;
the sum is 1.
= a/2, fateor
acumen et veritatem
animadversionis
ture.”
-G. Grandi 1133)