Results 1 to 3 of 3

Thread: ratio test for convergence

  1. #1
    Junior Member
    Joined
    Oct 2008
    Posts
    70

    ratio test for convergence

    I'm doing some review of series and using the ratio test for convergence.
    I have the problem (the summation of) n!x^n/n^n
    I remember the ratio of convergence = e but I forgot the trick to get to there. Would someone mind refreshing me?
    Thanks!
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor harish21's Avatar
    Joined
    Feb 2010
    From
    Dirty South
    Posts
    1,036
    Thanks
    10
    Quote Originally Posted by morganfor View Post
    I'm doing some review of series and using the ratio test for convergence.
    I have the problem (the summation of) n!x^n/n^n
    I remember the ratio of convergence = e but I forgot the trick to get to there. Would someone mind refreshing me?
    Thanks!
    for a series \sum_{n=0}^{\infty} {a_n} you have,

    X = lim_(n \rightarrow \infty) |\frac{a_{n+1}}{a_n}|

    Ratio test will tell you that:

    if X<1, the series is absolute convergent
    if X>1, the series diverges
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Super Member Deadstar's Avatar
    Joined
    Oct 2007
    Posts
    722
    Quote Originally Posted by morganfor View Post
    I'm doing some review of series and using the ratio test for convergence.
    I have the problem (the summation of) n!x^n/n^n
    I remember the ratio of convergence = e but I forgot the trick to get to there. Would someone mind refreshing me?
    Thanks!
    Ratio test -> \lim_{n \to \infty} \bigg{|}\frac{a_{n+1}}{a_n}\bigg{|}

    So we get...

    \lim_{n \to \infty}\bigg{|}\frac{(n+1)!x^{n+1}n^n}{n! x^n (n+1)^{n+1}}\bigg{|}

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

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

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

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


    = |x e^{-1}|
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Series Convergence Question i.e. Ratio Test
    Posted in the Calculus Forum
    Replies: 4
    Last Post: Nov 3rd 2010, 07:05 AM
  2. Ratio Test of Convergence
    Posted in the Calculus Forum
    Replies: 2
    Last Post: Feb 26th 2010, 02:37 PM
  3. Interval of convergence and ratio test
    Posted in the Calculus Forum
    Replies: 3
    Last Post: Apr 16th 2009, 06:15 AM
  4. Absolute Convergence and the Ratio Test
    Posted in the Calculus Forum
    Replies: 1
    Last Post: Nov 6th 2008, 12:06 PM
  5. Using ratio test to find radius of convergence
    Posted in the Calculus Forum
    Replies: 2
    Last Post: Jul 9th 2008, 07:00 AM

Search Tags


/mathhelpforum @mathhelpforum