Concrete mathematics : a foundation for computer science

572 ANSWERS TO EXERCISES
9.42 The hint follows since (,“,)/(z) = & $ a < &. Let
m = lcxn] = om ~ E. Then
<
n
1+i~+(&)2+...) = (;)S.
( m>(
so 1 ksa,, (;) = (:)0(l), an .d i tremains
to estimate (z). By Stirling’s approximation
we have In (z) =I -i 1nn-(an-e)ln(K-e/n)-((l--0()n+c) x
ln(l-cx+c/n)+0(1)=-~lnn-omlna-(1-ol)nIn(l-cx)+0(1).
9.43 The denominator has factors of the form z - w, where w is a complex
root of unity. Only the factor z - 1 occurs with multiplicity 5. Therefore
by (7.31), only one of the roots has a coefficient n(n4), and the coefficient is
c =5/(5!~1~5~10~25~50)=1/1500000.
9.44 Stirling’s approximation says that ln(xP”x!/(x-a)!) has an asymptotic
series
-a-(x+i-a)ln(l-a/x)-&(x ‘-(x-o())‘)
- &(x 3 - (x - cc) “) -’
in which each coefficient of xm~k is a polynomial in (x. Hence x “x!/(x - CX)! =
Co(R) +c1(a)x ’ + ... + c,(tx)xpn + 0(x-” ‘) as x + 03, where c,,(a) is a
polynomial in 01. We know that c, ( LX) = [,*,I (-1)" whenever 01 is an integer,
and LA1 is a polynomial in 01 of degree 2n; hence c, ( CX) = [ &*,,I (-1)” for
all real 01. In other words, the asymptotic formulas
generalize equations (6.13) and (6.11), which hold in the all-integer case.
9.45 Let the partial quotients of LX be (a,, al,. . . ), and let cc,,, be the continued
fraction l/(a, + CX,,~,) for m 3 1. Then D(cx,n) = D(cxl,n) <
D(olr, LarnJ) + al +3 < D(tx3, LcxzlcxlnJj) + al + a2 $6 < ... < D(Lx,+I,
~~m~...~~,n~...~~)+a~+~..+a,+3m