Mathematics for Computer Science

“mcs” — 2017/3/3 — 11:21 — page 590 — #598
590
Chapter 14
Sums and Asymptotics
Problem 14.26.
Determine which of these choices
‚.n/; ‚.n 2 log n/; ‚.n 2 /; ‚.1/; ‚.2 n /; ‚.2 n ln n /; none of these
describes each function’s asymptotic behavior. Full proofs are not required, but
briefly explain your answers.
(a)
n C ln n C .ln n/ 2
(b)
(c)
(d)
n 2 C 2n 3
n 2 7
nX
iD0
2 2iC1
ln.n 2 Š/
(e)
nX
kD1
k 1
1
2 k
Problem 14.27. (a) Either prove or disprove each of the following statements.
nŠ D O..n C 1/Š/
.n C 1/Š D O.nŠ/
nŠ D ‚..n C 1/Š/
nŠ D o..n C 1/Š/
.n C 1/Š D o.nŠ/
(b) Show that
n
3
nCe D o.nŠ/.
Problem 14.28.
Prove that
nX
k 6 D ‚.n 7 /:
kD1