Concrete mathematics : a foundation for computer science

We started with a
t in the recurrence,
and worked
hard to get rid of it.
But then after applying
a summation
factor, we came up
with another t.
Are sums good, or
bad, or what?
But your spelling is
a/wrong.
The solution, according to (2.10), is therefore
C, = 2(n + 1) f 1.
k=l k+l
2.2 SUMS AND RECURRENCES 29
The sum that remains is very similar to a quantity that arises frequently
in applications. It arises so often, in fact, that we give it a special name and
a special notation:
H, = ,+;+...+; r f;.
k=l
(2.13)
The letter H stands for “harmonic”; H, is a harmonic number, so called
because the kth harmonic produced by a violin string is the fundamental
tone produced by a string that is l/k times as long.
We can complete our study of the quicksort recurrence (2.12) by putting
C, into closed form; this will be possible if we can express C, in terms of H,.
The sum in our formula for C, is
We can relate this to H, without much difficulty by changing k to k - 1 and
revising the boundary conditions:
t 1
= ( l