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Math Help - Big O notation.Show using the definition of O(f)?

  1. #1
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    Big O notation.Show using the definition of O(f)?

    Hi I'm having some trouble with big O notation. I know how to do the simple ones but this is above my head. Any help would be appreciated.

    Show using only the definition of O(f) that:
    1) 5^n is O(n!)
    2)n! is not O(5^n)

    I know that 5^n<=n! for n=12 but I don't know how to show this. The book says to use induction 'but I don't know how and for 2 as well. Any ideas?

    Thanks in advance.
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  2. #2
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    1) 5^n is O(n!)
    Intuitive idea: in 5^n, all factors are equal to 5, whereas in n!, the factors start with 1 but soon overtake 5 and continue to increase. So, 5^5>5!, but this won't last long. Calculate 5^n and n! for n=6,7,...,12. As you said, 5^{12} is already less than 12!. This can be established by direct calculation (e.g., in Windows' calculator).

    Now what happens when we move from n to n+1 for n\ge 12? 5^n is multiplied by 5 to get 5^{n+1}; but n! is multiplied by n+1>5 to get (n+1)!. So, it is clear that 5^n<n! implies 5^{n+1}<(n+1)! for n\ge12.
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