AP 01 - Jan 11 - f ( x ) and g ( x ) be functions such that...

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Math 2602 Spring 2007 Additional Problems 1 Limit of a sequence, Rates of Growth Thursday, Jan 11th 1. Let ( a n ) be the sequence defined recursively by a n +3 = a n +2 + a n +1 + a n 3 , a 0 = a, a 1 = b, a 2 = c. Show that lim n →∞ a n = a + 2 b + 3 b 6 . 2. Show that the sequence a n = ± 1 + 1 n ² n is increasing and bounded above. 3. We build an exponential tower 2 2 2 2 . . . by defining a 0 = 1 and a n +1 = ( 2) a n for n N . Show that a n is monotonically increasing and bounded above by 2. What is the limit of a n ? 4. Let f ( x ) = (log x ) x and g ( x ) = x (log x ) . Determine whether f ( x ) = O ( g ( x )) or g ( x ) = O ( f ( x )) or both are true. Note that lim x →∞ x (log x ) (log x ) x = 0 5. Let
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Unformatted text preview: f ( x ) and g ( x ) be functions such that f ( x ) ≤ g ( x ) for all values of x ∈ R . For each of the following statements, write ALWAYS if the statement is always true, SOME-TIMES if the statement could be true (but requires more information than is given) or NEVER if the statement can never be true. Support your answer with an example. • f ( x ) = O ( g ( x )) • g ( x ) = O ( f ( x )) • f ( x ) = o ( g ( x )) • g ( x ) = o ( f ( x )) 1...
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This note was uploaded on 05/12/2010 for the course MATH 2602 taught by Professor Costello during the Spring '09 term at Georgia Institute of Technology.

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