Results 1 to 4 of 4

Math Help - A proof involving logs

  1. #1
    Junior Member
    Joined
    Apr 2010
    Posts
    30

    A proof involving logs

    Can anyone show that:

    log(N-1)! \leq \int_1^N \! log(x) \, dx \leq log(N)!
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Senior Member
    Joined
    Feb 2010
    Posts
    422
    \log (n!) = \log n(n-1)(n-2)\cdots2\cdot1 = \log n + \log (n-1) + \cdots + \log 2. This is a Riemann sum for the integral \int \log x. Can you show that the Riemann sum on the left of your inequality is less than the integral?
    Follow Math Help Forum on Facebook and Google+

  3. #3
    MHF Contributor

    Joined
    Apr 2005
    Posts
    15,368
    Thanks
    1313
    What Maddas is doing is setting up a Riemann sum for \int_1^N log(x)dx using the integers as the "break points"- that is \delta x= 1. Since log(x) is an increasing function, you get the "lower sum" using the left endpoints of each interval and the "upper sum" using the right endpoints.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Junior Member
    Joined
    Apr 2010
    Posts
    30
    So i would say that the one on the LHS is the Riemann integral for the limit 1 to N-1....

    So the middle bit of the inequality is the Riemann integral of the LHS + log(N)?
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 9
    Last Post: February 22nd 2011, 05:39 PM
  2. A proof involving logs
    Posted in the Calculus Forum
    Replies: 2
    Last Post: April 26th 2010, 03:23 PM
  3. Proofs involving logs
    Posted in the Calculus Forum
    Replies: 2
    Last Post: April 26th 2010, 05:36 AM
  4. limit involving logs
    Posted in the Calculus Forum
    Replies: 2
    Last Post: May 24th 2007, 05:31 PM
  5. Integration involving logs
    Posted in the Calculus Forum
    Replies: 1
    Last Post: May 7th 2007, 06:55 PM

Search Tags


/mathhelpforum @mathhelpforum