Concrete mathematics : a foundation for computer science

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9 EXERCISES 477
Suppose a, = O(f(n)) and b, = O(f(n)). Prove or disprove that the
convolution ~~==, akb+k is also O(f(n)), in the following cases:
a f(n) = n-“:, a > 1.
b f(n) = a-n,, OL > 1.
Prove (9.1) and (9.2), with which we opened this chapter.
Equation (9.~1) shows how to evaluate In lo! with an absolute error <
A. Therefore if we take exponentials, we get lo! with a relative
error that is less than e1/126000000 - 1 < 1 Op8. (In fact, the approximation
gives 3628799.9714.) If we now round to the nearest integer, knowing that
lo! is an integer, we get an exact result.
Is it always possible to calculate n! in a similar way, if enough terms of
Stirling’s approximation are computed? Estimate the value of m that
gives the best approximation to Inn!, when n is a fixed (large) integer.
Compare the absolute error in this approximation with n! itself.
Use Euler’s summation formula to find the asymptotic value of Hipa) =
,E:=, ka, where a is any fixed real number. (Your answer may involve a
constant that you do not know in closed form.)
Exercise 5.13 defines the hyperfactorial function Q,, = 1 122 . . . nn. Find
the asymptotic value of Qn with relative error O(n’). (Your answer
may involve a constant that you do not know in closed form.)
Estimate the function 1”’ 2’i2 . . . n”” as in the previous exercise.
Find the asymptotic value of &O k’epkL”’ with absolute error O(n3),
when 1 is a fixed nonnegative integer.
Evaluate tk20 l/(ck+cm) with absolute error O(C-~~), when c > 1 and
m is a positive iinteger.
Exam problems
32 Evaluate eHn+Hiz’ with absolute error O(n-‘).
33 Evaluate tkao (;)/n’ with absolute error O(np3).
34 Determine values A through F such that (1 + 1 /n)“Hn is
E(lnn)’ Flnn
An+B(lnn)2+Clnn+D+I+-+O(n-‘).
35 Evaluate I:=, 1 /kHk with absolute error 0( 1).
36 Evaluate S, = xF=, l/(n’ + k2) with absolute error O(n5).
37 Evaluate IF=, ln mod k) with absolute error O(nlogn).
38 Evaluate tkaO kk (i) with relative error 0 (n-’ ) .