Concrete mathematics : a foundation for computer science

9.6 FINAL SUMMATIONS 475
is bounded by 2n times its largest term, which occurs at the cutoff point k z
nl/z+E. This term is known to be approximately bk (n), which is exponentially
small compared with A,; and an exponentially small multiplier wipes out the
factor of 2n.
Thus we have ;successfully applied the tail-exchange trick to prove the
estimate
22n zz if 0 < E < 4. (9.99)
Thanks for reading We may choose e = i and conclude that
this, hope it comes
in handy.
-The authors
0 = tln27r.
QED.
Exercises
Warmups
1 Prove or disprove: If f 1 (n) 4 gl (n) and f2 (n) + g2 (n), then we have
fl (n) + f2(n) -: 91 (n) + 92(n).
2 Which function grows faster:
a n(inn) or (Inn)n?
b n(lnln’nn) or (Inn)!?
c (n!)! or I((n - l)!)! (n - l)!“!?
d FfH,, or HF,?
3 What’s wrong with the following argument? “Since n = O(n) and 2n =
O(n) and so on, we have XL=:=, kn = Et=, O(n) = O(n2).”
4 Give an example of a valid equation that has O-notation on the left but
not on the right. (Do not use the trick of multiplying by zero; that’s too
easy.) Hint: Consider taking limits.
5 Prove or disprove: O(f(n) + g(n)) = f(n) + O(g(n)), if f(n) and g(n)
are positive for all n. (Compare with (g.zT).)
6 Multiply (lnn+y+O(l/n)) by (n+O(fi)), and express your answer
in O-notation.
7 Estimate xkaO eek/” with absolute error O(n-’ ).
Basics
8 Give an example of functions f(n) and g(n) such that none of the three
relations f(n) 4 g(n), f(n) + g(n), f(n) x g(n) is valid, although f(n)
and g(n) both. increase monotonically to 03.