Results 1 to 2 of 2

Math Help - sequences with factor'ials?

  1. #1
    Junior Member
    Joined
    May 2009
    Posts
    71

    sequences with factor'ials?

    confirm is this one divergent?

    inf x!
    sum ------------------------
    x=1 1 * 3 * 5 ... ( 2 x -1)

    I said that we should start by replacing all n with n+1

    Then we multiply an+1 by the reciprocal of an

    That leads me to a messy fraction.

    I get all messed up with a fraction that looks like:

    (x+1)(2x-1)
    -----------
    2x+1

    I come up with the series is divergent but due to abs value of ratio an+1 to an.

    How do you solve this so the fraction obtained after the reciprocal is one that is easily taken a limit of to determine divergence or convergence?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Super Member Deadstar's Avatar
    Joined
    Oct 2007
    Posts
    722
    Quote Originally Posted by orendacl View Post
    confirm is this one divergent?

    inf x!
    sum ------------------------
    x=1 1 * 3 * 5 ... ( 2 x -1)

    I said that we should start by replacing all n with n+1

    Then we multiply an+1 by the reciprocal of an

    That leads me to a messy fraction.

    I get all messed up with a fraction that looks like:

    (x+1)(2x-1)
    -----------
    2x+1

    I come up with the series is divergent but due to abs value of ratio an+1 to an.

    How do you solve this so the fraction obtained after the reciprocal is one that is easily taken a limit of to determine divergence or convergence?
    Ratio test.

    \lim_{n \to \infty}\bigg{|} \frac{a_{n+1}}{a_n} \bigg{|}

    = \lim_{n \to \infty}\bigg{|} \frac{(x+1)! \cdot (1 \cdot 3 \cdot 5 \cdot \dots \cdot (2x-1)}{x! \cdot (1 \cdot 3 \cdot 5 \cdot \dots \cdot (2x+1)} \bigg{|}

    = \lim_{n \to \infty}\bigg{|} \frac{x+1}{2x(2x+1)}  \bigg{|}

    = \lim_{n \to \infty}\bigg{|} \frac{x+1}{4x^2 + 2x}  \bigg{|} < 1. Hence converges.

    In fact, I think it converges to...

    1 + \frac{\pi}{2}...
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 0
    Last Post: July 19th 2011, 11:45 AM
  2. factor analysis , extract second factor loading
    Posted in the Advanced Statistics Forum
    Replies: 0
    Last Post: June 1st 2011, 05:17 AM
  3. factor analysis , extract second factor loading
    Posted in the Differential Geometry Forum
    Replies: 0
    Last Post: May 30th 2011, 05:29 AM
  4. Monotone sequences and Cauchy sequences
    Posted in the Differential Geometry Forum
    Replies: 3
    Last Post: March 21st 2009, 08:59 PM
  5. Replies: 5
    Last Post: January 16th 2008, 04:51 PM

Search Tags


/mathhelpforum @mathhelpforum