Concrete mathematics : a foundation for computer science

The laws of the
jungle.
27 Compute A(cx), and use it to deduce the value of xE=, (-2)k/k.
28 At what point does the following derivation go astray?
Exam problems
F[j=k+l]-k[j=k-1] ==( k>l j31
=
=(
j>l k>l
=x(
29 Evaluate the sum ,& (-l)kk/(4k2 - 1).
;[j=k+l]-k[j=k-1]
;[k=j-l]-i[k=j+l]
2 EXERCISES 65
j-l
- -
j
-
i31 i j+l = && = -'.
30 Cribbage players have long been aware that 15 = 7 + 8 = 4 + 5 + 6 =
1 + 2 + 3 + 4 + 5. Find the number of ways to represent 1050 as a sum of
consecutive positive integers. (The trivial representation ‘1050’ by itself
counts as one way; thus there are four, not three, ways to represent 15
as a sum of consecutive positive integers. Incidentally, a knowledge of
cribbage rules is of no use in this problem.)
31 Riemann’s zeta function c(k) is defined to be the infinite sum
Prove that tka2(L(k) - 1) = 1. What is the value of tk?l (L(2k) - l)?
32 Let a 2 b = max(0, a - b). Prove that
tmin(k,x’k) = x(x: (2k+ 1))
k>O k?O
for all real x 3 0, and evaluate the sums in closed form.
Bonus problems
33 Let /\kcK ok denote the minimum of the numbers ok (or their greatest
lower bound, if K is infinite), assuming that each ok is either real or foe.
What laws are valid for A-notation, analogous to those that work for t
and n? (See exercise 25.)
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