Results 1 to 4 of 4

Math Help - Little o notation.

  1. #1
    MHF Contributor Also sprach Zarathustra's Avatar
    Joined
    Dec 2009
    From
    Russia
    Posts
    1,506
    Thanks
    1

    Little o notation.

    Prove that:

    \frac{(1+\frac{1}{n})^x}{1+\frac{x}{n}}=1+\frac{x(  x-1)}{2n^2}+o\left(\frac{1}{n^2}\right)
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor
    Joined
    Oct 2009
    Posts
    5,559
    Thanks
    785

    Re: Little o notation.

    I believe that (1+y)^\alpha=1+\alpha y+\alpha(\alpha-1)y^2/2+o(y^2) (perhaps the o(y^2) part needs to be double-checked). Expanding the nominator and the denominator in this way gives you the right-hand side.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Grand Panjandrum
    Joined
    Nov 2005
    From
    someplace
    Posts
    14,972
    Thanks
    4

    Re: Little o notation.

    Quote Originally Posted by Also sprach Zarathustra View Post
    Prove that:

    \frac{(1+\frac{1}{n})^x}{1+\frac{x}{n}}=1+\frac{x(  x-1)}{2n^2}+o\left(\frac{1}{n^2}\right)
    You are interested in the rate of growth of the left hand side as n \to \infty so put \varepsilon=1/n and expand as a Mclaurin series with the remainder term after the term in \varepsilon^2. Changing back to n will give the result.

    CB
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor Also sprach Zarathustra's Avatar
    Joined
    Dec 2009
    From
    Russia
    Posts
    1,506
    Thanks
    1

    Re: Little o notation.

    Quote Originally Posted by CaptainBlack View Post
    You are interested in the rate of growth of the left hand side as n \to \infty so put \varepsilon=1/n and expand as a Mclaurin series with the remainder term after the term in \varepsilon^2. Changing back to n will give the result.

    CB


    It worked! Thank you!
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 1
    Last Post: March 10th 2011, 03:23 AM
  2. nCr notation
    Posted in the Discrete Math Forum
    Replies: 2
    Last Post: August 30th 2010, 11:51 AM
  3. Notation
    Posted in the Discrete Math Forum
    Replies: 3
    Last Post: June 4th 2010, 12:36 PM
  4. Replies: 1
    Last Post: August 26th 2009, 01:40 PM
  5. Notation
    Posted in the Advanced Statistics Forum
    Replies: 2
    Last Post: March 27th 2009, 09:26 PM

Search Tags


/mathhelpforum @mathhelpforum