mid1review_some_solns

mid1review_some_solns - CMPS 101 Midterm 1 Review Solutions...

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CMPS 101 Midterm 1 Review Solutions to selected problems Problem 2 State whether the following assertions are true or false. If any statements are false, give a related statement which is true. a. )) ( ( ) ( n g O n f = implies )) ( ( ) ( n g o n f = . False )) ( ( ) ( n g o n f = implies )) ( ( ) ( n g O n f = b. )) ( ( ) ( n g O n f = if and only if )) ( ( ) ( n f n g Ω = . True c. )) ( ( ) ( n g n f Θ = if and only if L n g n f n = )) ( / ) ( ( lim , where < < L 0 . False < < L 0 and L n g n f n = )) ( / ) ( ( lim implies )) ( ( ) ( n g n f Θ = Problem 3 Prove that )) ( ) ( ( )) ( ( )) ( ( n g n f n g n f Θ = Θ Θ . In other words, if )) ( ( ) ( 1 n f n h Θ = and )) ( ( ) ( 2 n g n h Θ = , then )) ( ) ( ( ) ( ) ( 2 1 n g n f n h n h Θ = . Proof: By hypothesis there exist positive constants 1 n , 2 n , 1 a , 1 b , 2 a , and 2 b such that ) ( ) ( ) ( 0 : 1 1 1 1 n f b n h n f a n n and ) ( ) ( ) ( 0 : 2 2 2 2 n g b n h n g a n n If ) , max( 2 1 0 n n n n = , then both inequalities hold. Let 2 1 a a c = , and 2 1 b b d = . Since everything in sight is non-negative, we can multiply these two inequalities to get ) ( ) ( ) ( ) ( ) ( ) ( 0 : 2 1 0 n g n f d n h n h n g n f c n n , and hence )) ( ) ( ( ) ( ) ( 2 1 n g n f n h n h Θ = as required. /// Problem 4 Let ) ( n f and ) ( n g be asymptotically positive functions (i.e. 0 ) ( > n f and 0 ) ( > n g for all sufficiently large n ), and suppose that )) ( ( ) ( n g n f Θ = . Does it necessarily follow that Θ = ) ( 1 ) ( 1 n g n f ? Either prove this statement, or give a counter-example.
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