Concrete mathematics : a foundation for computer science

476 ASYMPTOTIC23
9 Prove (9.22) rigorously by showing that the left side is a subset of the
right side, according to the set-of-functions definition of 0.
10 Prove or disprove: cos O(x) = 1 + 0(x2) for all real x.
11 Prove or disprove: 0(x-t y)2 = 0(x2) + O(y2).
12 Prove that
asn-koc.
1 + f + O(np2) = (1 + f)(l + O(nm2)),
13 Evaluate (n + 2 + O(n I))” with relative error O(n ‘).
14 Prove that (n + a)“+P � = nn+pea(l + ol(@ -- tcY)n-’ + O(n2)).
15 Give an asymptotic form.ula for the “middle” trinomial coefficient (n3cn),
3 ,
correct to relative error O(n3).
16 Show that if B(l -x) = -B(x) 3 0 for 0 < x < i, we have
s
b
B(H) f(x) dx 3 0
a
if we assume also that f”(x) 3 0 for a < x c; b.
17 Use generating functions to show that B,(i) = (2’-“’ - l)B,, for all
m 3 0.
18 Find tk (2r)(x with relative error O(np’/4), when a > 0.
Homework exercises
19 Use a computer to compare the left and right sides of the approximations
in Table 438, when n = 10, z = a = 0.1, and O(f(n)) = O(f(z)) = 0.
20 Prove or disprove the following estimates, as n + 00:
a o( (&J2) = O(lJ;;12)~
b e(‘+oc’/n))z c e + (0(1/n).
c n! = O(((1 - l/n]nn)n).
21 Equation (9.48) gives the nth prime with relative error O(logn))2. Improve
the relative error to O(logn))3 by starting with another term of
(9.31) in (9.46).
22 Improve (9.54) to O(np3).
23 Push the approximation (9.62) further, getting absolute error O(n--3).
Hint: Let g,, = c/(n + 1) (n + 2) + h,; what recurrence does h,, satisfy?