Question

Problem 3.29. Complete the proof of the smoothness rule (end of Section 3.4) by providing details of the proof that t(ne 2(f(n)) whenever f(n) is a smooth func- tion and t(n) is an eventually nondecreasing function such that t(ne (f(n) n is a power of b). Note that the smoothness rule applies equally well if we replace by 0 or by 12 in its statement. This is because your proof that t(n)e (f(n)) does not use the fact that t(n)e Of(n) | n is a power of b), just as the proof we gave in Section 3.4 that tine (f(n)) did not use the fact that tinde s2(f(n) in is a power of b). Hint: This is very similar to the proof that t(n)€ O(f(n)) under the same condi- tions. Use the fact that if f(bm)scf(m) then f(m)2c-1f(bm).

Answer 1

Complete the proof of the Smoothness srule lend of Seetion 3.4) by providing detaills of the proof that (n2 Go (f(n)) whenever f(n)is a smooth function and tom is an eventually non decreasing function such that tr] EO (H)in is a power of b). to Ans dimoothness rule: İt tro co(min-bk) and t is Smooth and to is eventually non-decreasing, then tm e elf (1) un conditionelly. Foo. Ex : If to? E o (2/a-ak) and eventually non- decreasing ther towe O (12) laconditionally. . If. How is Smooth function for some integeon value, let us say some bal then it is Smooth for all such b's. +62 c-ct (bn) and tol st(n+1) which is. fin)ç ef (an) Ex:= () = n3 is Smooth because f@co2 = Cam =8120m3) Now b(9) can is not Smooth becouse hamba gan+() So their fore if we find for some bal that .. tim anh (f(n) where is a power of b then. t.(n) is a smooth function, we have tm-offm) for all. n.