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Homework Help: Finding liminf of p_n/n where p_n = nth prime

  1. Mar 2, 2007 #1
    1. The problem statement, all variables and given/known data

    Find the lim inf of p_n/n where p_n is the nth prime.


    2. Relevant equations

    Well p_n ~ n logn, but I'm not sure if a simple substitution would work. This question may be incredibly trivial or open, and I can't figure out which.

    I'm also wondering if the sequence above is monotone decreasing for sufficiently large n (this is not true for small n).
     
    Last edited: Mar 2, 2007
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  3. Mar 3, 2007 #2

    Gib Z

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    You have to think what [tex]P_n [/tex]~ [tex]n \log n[/tex] actually means!

    It means the quotient of the 2 functions as n approaches infinity is 1, note this does not mean the difference of the functions as n approaches infinity is 0.

    eg n+2~n, but the difference of the functions will always be 2.

    Anywho, so we know [tex]\lim_{n\rightarrow {\infty}} \frac{P_n}{n\log n} = 1[/tex].

    Since we want [tex]\lim_{n\rightarrow {\infty}} \frac{P_n}{n}[/tex], we can make the simple substitution and get log n.
     
    Last edited: Mar 3, 2007
  4. Mar 3, 2007 #3
    Ok I wrote that backwards. I want n/p_n, not p_n/n. The question is whether lim n/(nlog n)=0 implies that lim inf n/p_n=0. More importantly is whether n/p_n is monotone for large n.
     
  5. Mar 4, 2007 #4

    Gib Z

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    Well as per my previous post, the substitution is justified. So Your first question is true, it implies it. The 2nd part, I would not know.
     
  6. Mar 4, 2007 #5
    An equivalent question is whether [tex]np_{n+1}-(n+1)p_n<0[/tex] for only finitely many n.
     
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