Concrete mathematics : a foundation for computer science

2.7 INFINITE SUMS 59
We might also try setting x = -1 in the formula &O xk = 1 /(l - x),
since we’ve proved that this formula holds when 0 < x < 1; but then we are
forced to conclude that the infinite sum is i, although it’s a sum of integers!
Another interesting example is the doubly infinite tk ok where ok =
l/(k+ 1) for k 3 0 and ok = l/(k- 1) for k < 0. We can write this as
.'.+(-$)+(-f)+(-;)+l+;+f+;+'.'. (2.58)
If we evaluate this sum by starting at the “center” element and working
outward,
..+ (-$+(-f +(-; +(l)+ ;,+ g-t ;> +...,
we get the value 1; and we obtain the same value 1 if we shift all the parentheses
one step to the left,
+(-j+(-;+cf+i-;)+l)+;)+:)+.y
because the sum of all numbers inside the innermost n parentheses is
-----...-
1 1 1
j+,+;+...+L = l-L_ 1
nfl n n - l n K-3’
A similar argument shows that the value is 1 if these parentheses are shifted
any fixed amount to the left or right; this encourages us to believe that the
sum is indeed 1. On the other hand, if we group terms in the following way,
..+(-i+(-f+(-;+l+;,+f+;)+;+;)+...,
the nth pair of parentheses from inside out contains the numbers
1 1 1
---- -...- 2+,+;+...+
n+l n
& + & = 1 + Hz,, - &+I .
We’ll prove in Chapter 9 that lim,,,(Hz,-H,+, ) = ln2; hence this grouping
suggests that the doubly infinite sum should really be equal to 1 + ln2.
There’s something flaky about a sum that gives different values when
its terms are added up in different ways. Advanced texts on analysis have
a variety of definitions by which meaningful values can be assigned to such
pathological sums; but if we adopt those definitions, we cannot operate with
x-notation as freely as we have been doing. We don’t need the delicate refinements
of “conditional convergence” for the purposes of this book; therefore
Is this the first page we’ll stick to a definition of infinite sums that preserves the validity of all the
with no graffiti? operations we’ve been doing in this chapter.