Concrete mathematics : a foundation for computer science

54 SUMS
This formula indicates why harmonic numbers tend to pop up in the solutions
to discrete problems like the analysis of quicksort, just as so-called natural
logarithms arise naturally in the solutions to continuous problems.
Now that we’ve found an analog for lnx, let’s see if there’s one for e’.
What function f(x) has the property that Af(x) = f(x), corresponding to the
identity De” = e”? Easy:
f(x+l)-f(X) = f(x) w f(x+ 1) = 2f(x);
so we’re dealing with a simple recurrence, and we can take f(x) = 2” as the
discrete exponential function.
The difference of cx is also quite simple, for arbitrary c, namely
A(?) = cx+’ - cX = (c - 1)~“.
Hence the anti-difference of cx is c’/(c - 1 ), if c # 1. This fact, together with
the fundamental laws (2.47) and (2.48), gives us a tidy way to understand the
general formula for the sum of a geometric progression:
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