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**Unformatted text preview: **Solutions for Homework 1 f(n) log k n n k p n log( n !) g(n) n c n n sinn log( n n ) Is f ( n ) O ( g ( n ))? Yes Yes No Yes Is f ( n ) ( g ( n ))? No No No Yes Is f ( n ) ( g ( n ))? No No No Yes Is f ( n ) o ( g ( n ))? Yes Yes No No Table 1: Problem 1: (Grade 16 pts): In Table 1, k 1 ; > ; c > 1 : Please answer yes or no, and also justify your answer in each case. Column1: We will prove that log k n = o( n ). We want to show that for any constant c > 0, there exists a constant n > 0 : log k n < cn : Take the log of both sides log log k n < log cn k log log n < log c + log n This is true for large enough n (it can be proven using L'Hopital's rule). So, log k n = o( n ). (1) From (1), we conclude that: log k n = O ( n ) : (2) log k n 6 = ( n ) : (3) log k n 6 = ( n ) : (4) Column2: We will prove that n k = o ( c n ). We want to show that for any constant a > 0, there exists a constant n > 0 : n k < ac n log n k < log ac n k log n < log a + log cn This is true for large enough n (it can be proven using L'Hopital's rule). So, n k = o ( c n ). (5) 1 From (5), we conclude that: n k = O ( c n ) : (6) n k 6 = ( c n ) : (7) n k 6 = ( c n ) : (8) Column3: Observe the function n sinn : The value of the exponent is oscillating between 1 and -1, taking all values in between.The value of the exponent is oscillating between 1 and -1, taking all values in between....

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