1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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
  2. jcsd
  3. Mar 3, 2007 #2

    Gib Z

    User Avatar
    Homework Helper

    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

    User Avatar
    Homework Helper

    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.
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook