Mathematics for Computer Science
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“mcs” — 2017/3/3 — 11:21 — page 589 — #597<br />
14.7. Asymptotic Notation 589<br />
8. 2 2100<br />
9. 2 2n<br />
10. 2 n<br />
11. 3 n<br />
12. n2 n<br />
13. 2 nC1<br />
14. 2n<br />
15. 3n<br />
16. log .nŠ/<br />
17. log 2 n<br />
18. log 10 n<br />
19. 2:1 p n<br />
20. 2 2n<br />
21. 4 n<br />
22. n 64<br />
23. n 65<br />
24. n n<br />
Problem 14.25.<br />
Let f , g be nonnegative real-valued functions such that lim x!1 f .x/ D 1 and<br />
f g.<br />
(a) Give an example of f; g such that NOT.2 f 2 g /.<br />
(b) Prove that log f log g.<br />
(c) Use Stirling’s <strong>for</strong>mula to prove that in fact<br />
log.nŠ/ n log n
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