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Math Help - Θ-classes to arrange functions?

  1. #1
    Junior Member
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    Θ-classes to arrange functions?

    Use the rules and definition for ordering Θ-classes to arrange the following in order from lowest to highest:
    lg n ; (lg n)^6 ; lg(n^n) ; n^4 ; n^(1/2) ; 5n^3+n^2 ; 1^n ; (1.001)^n ; 5^n

    I used the rules as I understood them and ended very uncertainly with the following order:

    1^n (1 to the power of anything is)
    lg n
    n^(1/2)
    lg (n^n)
    5n^3+n^2
    n^4
    (lg n)^6
    (1.001)^n
    5^n

    I'm really not sure about the order so if someone could please check and tell me where and why I went wrong I would really appreciate it.

    Thanks in advance.
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  2. #2
    MHF Contributor
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    The general rule is that for every a_1,a_2>1 and \beta_1,\beta_2,\beta_3>0, for the functions f_1(n)=(\log_{a_1} n)^{\beta_1}, f_2(n)=n^{\beta_2} and f_3(n)={a_2}^{{\beta_3} n}, we have \lim_{n\to\infty}f_1(n)/f_2(n)=0 and \lim_{n\to\infty}f_2(n)/f_3(n)=0. So you answer is correct except for the place of (\lg n)^6.

    By the way, did you realize that for n=1000\,000, n^4=10^{24} (the order of the Earth's mass in kilograms), while (\lg n)^6<50\,000 (decimal logarithm)?
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