Concrete mathematics : a foundation for computer science

0.577 exactly?
Maybe they mean
l/d.
Then again,
maybe not.
2.6 FINITE AND INFINITE CALCULUS 53
Now let’s make sure that the crucial difference property holds for our
newly defined falling powers. Does Ax2 = mx* when m < O? If m = -2,
for example, the difference is
A& =
1 1
(x+2)(x+3) - (x+1)(x+2)
(x+1)-(x+3)
= (x+1)(%+2)(x+3)
= -2y-3,
Yes -it works! A similar argument applies for all m < 0.
Therefore the summation property (2.50) holds for negative falling powers
as well as positive ones, as long as no division by zero occurs:
x
b
Xmfl b
x”& = - for mf-1
a m+l (1’
But what about when m = -l? Recall that for integration we use
sb
x-’ dx = lnx b
a
a
when m = -1. We’d like to have a finite analog of lnx; in other words, we
seek a function f(x) such that
x-' =
1
- = Af(x) = f(x+ 1)-f(x).
x+1
It’s not too hard to see that
f(x) = ; + ; f...f ;
is such a function, when x is an integer, and this quantity is just the harmonic
number H, of (2.13). Thus H, is the discrete analog of the continuous lnx.
(We will define H, for noninteger x in Chapter 6, but integer values are good
enough for present purposes. We’ll also see in Chapter 9 that, for large x, the
value of H, - In x is approximately 0.577 + 1/(2x). Hence H, and In x are not
only analogous, their values usually differ by less than 1.)
We can now give a complete description of the sums of falling powers:
b
z
a
ifmf-1;
x”6x = (2.53)
ifm=-1.