Results 1 to 3 of 3
Like Tree1Thanks
  • 1 Post By HallsofIvy

Math Help - Help needed solving an inequality involving the natural log

  1. #1
    Junior Member
    Joined
    Jul 2012
    From
    NOVA
    Posts
    25

    Help needed solving an inequality involving the natural log

    I'm trying to solve a simple alternative series question, but I cannot get past the following inequality to calculate my nth term for the required error:

    e^(-n-1).(n+1) < 0.01

    I solved it all the way to:

    ln n < (n + 1) + ln 0.01

    I can't figure out how to solve for n after this.

    Any help will be greatly appreciated,

    Giest
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor

    Joined
    Apr 2005
    Posts
    15,693
    Thanks
    1466

    Re: Help needed solving an inequality involving the natural log

    There is no elementary way to solve that. What we can do is let u= n+ 1 so the function can be written e^{-u}u. We can further let v= -u so that e^{-u}u= e^v(-v)= -e^vv. Now, the best way to solve such an inequality is to first solve the associate equation- here, e^{-(n+1)}(n+1)= -ve^v= 0.01 which is, of course, the same as ve^v= -0.01. And the solution to that is x= W(-0.01) where W is the "Lambert W function" (also known as the "omega function" or "product log" function) which is defined as the inverse function to f(x)= xe^x. Looking at a graph of y= xe^x we can see there are two roots for xe^x= -0.01, one at about -0.01 and the other at about -6.47. The inequality is satisfied for x below the root around -6 and above the root around -.01. (I used the "Newton-Raphson" iteration method to solve for those two values.)
    Last edited by HallsofIvy; July 25th 2012 at 05:09 PM.
    Thanks from Giestforlife
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Junior Member
    Joined
    Jul 2012
    From
    NOVA
    Posts
    25

    Re: Help needed solving an inequality involving the natural log

    Thanks for the help on this one.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Natural Numbers - help needed
    Posted in the Discrete Math Forum
    Replies: 1
    Last Post: March 4th 2010, 11:02 AM
  2. Solving for x in this natural log inequality
    Posted in the Pre-Calculus Forum
    Replies: 2
    Last Post: January 7th 2010, 05:34 PM
  3. derivative involving natural log
    Posted in the Calculus Forum
    Replies: 3
    Last Post: March 9th 2009, 08:54 PM
  4. [SOLVED] Inequality proof involving natural numbers.
    Posted in the Discrete Math Forum
    Replies: 2
    Last Post: February 13th 2009, 08:19 AM
  5. Proof needed for an inequality involving definite integrals
    Posted in the Advanced Math Topics Forum
    Replies: 0
    Last Post: February 10th 2009, 10:00 AM

Search Tags


/mathhelpforum @mathhelpforum