354 CHAPTER 9 CALCULUS
9.4. 19 (Putnam 1997) Evaluater[(x_X^3 +�_�+ ... )
Jo^2 2·4^2 ·4·^6x (^1 + �� + 2 t 42 + 22. �. 62 + .. -) ] dx.9.4.20 (Putnam 1990) Is there an infinite sequence
ao, a I , a2,. .. of nonzero real numbers such that for
n = 1,2,3, ... the polynomial
Pn(x) = ao +alx+a 2 ..? + ... +an�
has exactly n distinct real roots?
9.4.2 1 Prove that
Si;X =
Dcos(;),(a) using telescoping;
(b) using power series.
9.4.22 In Example 9.4.9, U(n,k) was defined by a
nasty integral. Directly evaluate this integral and show
that it equals 1/ (n + 1) using
(a) repeated applications of integration by parts;
(b) manipulation of the binomial series.Eulerian Mathematics and Number Theory
The following challenging problems are somewhat in
terrelated, all involving manipulations similar to Ex
ample 9.4.7. You may need to reread the combina
torics and number theory chapters, and some familiar
ity with probability is helpful for the last two prob
lems. (For the definition of " q" Jl, cr, see pages 162 ,
236, 238, 235, respectively.)
9.4.23 Evaluate
1 1 1 I
12 + 32 + 53 + 72 + ....
9.4.24 Let S be the set of integers whose prime factor
izations include only the primes 3, 5, and 7. Does the
sum of the reciprocals of the elements of S converge,
and if so, to what?
9.4.25 Consider the argument used in Example 9.4.7.
(a) Did this argument really require the funda
mental theorem of arithmetic (unique factoriza
tion)?
(b) Make this argument rigorous, by considering
only finite partial sums and products.pS
9.4.26 Show that '(s) = n ..-=--s.
p pnme P
9.4.27 Compute
('(2) -1)+('(3)-1)+('(4) -1)+···.
9.4.28 Let P = {4,8,9, 16,... } be the set of perfect
powers, i.e., the set of positive integers of the form ab,
where a and b are integers greater than I. Prove that
1
J�j-l=l
.9.4.29 Modify the argument used in Example 9.4.8 to
show that ,(4) = n4/90. Can you generalize this to
find a formula for '(2n)? How about ,(2n + I)?
9.4.30 Find a sequence n I, n 2 , ... such that9.4.3 1 Prove that9.4.32 Use the ideas from Problem 9.2.37 and Exam
ple 9.4.7 to show that, for any positive integer n, the
sum of the reciprocals of the prime numbers that do
not exceed n is greater than log(logn)) - 1/2. Use this
to show that 4 ..!.. diverges.
p pnme P
9.4.33 Fix a positive integer n. Let PI, ... , Pk be
the primes less than or equal to..;n. Let Qn := PI.
P 2 ··· Pk· Let n(x) denote the number of primes less
than or equal to x. Show thatn(n) = -I + n(..;n) + � Jl(d) l� J.
diQn
9.4.34 For n E N,x E R, define q,(n,x) to equal the
number of positive integers less than or equal to x that
are relatively prime to n. For example, q,(n,n) is just
plain old q,(n). Find a formula for computing q,(n,x).
9.4.35 Show that the number of pairs (x,y), where x
and y are relatively prime integers between 1 and n in
clusive, is9.4.36 Show that the probability that two randomly
chosen positive integers are relatively prime is
6
2.
n