Results 1 to 7 of 7

Math Help - convergence proof

  1. #1
    Member
    Joined
    Jun 2008
    Posts
    170

    convergence proof

    Prove that the sequence  a_{n} = \frac{n^3}{n!} converges.

    Intuitively this seems to converge to  0 (e.g. its  o(n) ). So for all  \varepsilon > 0 there is an  N \in \mathbb{N} , such that for  n \geq N ,  |a_n| < \varepsilon .

    Is that the end of the proof? Would you choose an  N \in \mathbb{N} such that  \frac{N^3}{N!} < \varepsilon ?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    is up to his old tricks again! Jhevon's Avatar
    Joined
    Feb 2007
    From
    New York, USA
    Posts
    11,663
    Thanks
    3
    Quote Originally Posted by particlejohn View Post
    Prove that the sequence  a_{n} = \frac{n^3}{n!} converges.

    Intuitively this seems to converge to  0 (e.g. its  o(n) ). So for all  \varepsilon > 0 there is an  N \in \mathbb{N} , such that for  n \geq N ,  |a_n| < \varepsilon .

    Is that the end of the proof? Would you choose an  N \in \mathbb{N} such that  \frac{N^3}{N!} < \varepsilon ?
    Consider the series \sum a_n. note that it converges by the ratio test. thus, \lim a_n = 0 by the theorem \sum b_n \text{ converges} \implies \lim b_n = 0 (i forgot what it's called ), and hence a_n converges (to 0).
    Follow Math Help Forum on Facebook and Google+

  3. #3
    MHF Contributor kalagota's Avatar
    Joined
    Oct 2007
    From
    Taguig City, Philippines
    Posts
    1,026
    Quote Originally Posted by Jhevon View Post
    C theorem \sum b_n \text{ converges} \implies \lim b_n = 0 (i forgot what it's called ),
    contrapositive of the divergence test..

    ratio test: \lim \frac{a_{n+1}}{a_n} = \lim\frac{(n+1)^3}{(n+1)!} \cdot \frac{n!}{n^3} = \lim\frac{(n+1)^3}{(n+1)} \cdot \frac{1}{n^3} = \lim\frac{(n+1)^2}{n^3} =0 < 1
    Last edited by kalagota; July 6th 2008 at 07:22 PM. Reason: i figure out there was a mistake on my computation..
    Follow Math Help Forum on Facebook and Google+

  4. #4
    is up to his old tricks again! Jhevon's Avatar
    Joined
    Feb 2007
    From
    New York, USA
    Posts
    11,663
    Thanks
    3
    Quote Originally Posted by kalagota View Post
    contrapositive of the divergence test..
    haha, i thought so. but somehow i figured mathematicians would have been creative enough to come up with a short name for it. i didn't want to say that whole phrase and sound archaic
    Follow Math Help Forum on Facebook and Google+

  5. #5
    MHF Contributor kalagota's Avatar
    Joined
    Oct 2007
    From
    Taguig City, Philippines
    Posts
    1,026
    Quote Originally Posted by Jhevon View Post
    haha, i thought so. but somehow i figured mathematicians would have been creative enough to come up with a short name for it. i didn't want to say that whole phrase and sound archaic
    It make sense..

    it was a remark of the divergence test when we discussed it before.. haha

    EDIT: OFF-TOPIC.. how to join your team?
    Follow Math Help Forum on Facebook and Google+

  6. #6
    is up to his old tricks again! Jhevon's Avatar
    Joined
    Feb 2007
    From
    New York, USA
    Posts
    11,663
    Thanks
    3
    Quote Originally Posted by kalagota View Post
    It make sense..

    it was a remark of the divergence test when we discussed it before.. haha
    haha, yup

    EDIT: OFF-TOPIC.. how to join your team?
    i answered you in a PM
    Follow Math Help Forum on Facebook and Google+

  7. #7
    MHF Contributor kalagota's Avatar
    Joined
    Oct 2007
    From
    Taguig City, Philippines
    Posts
    1,026
    Quote Originally Posted by Jhevon View Post
    i answered you in a PM
    yes, and i am fixing it already.. thanks!
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Convergence Proof
    Posted in the Differential Geometry Forum
    Replies: 2
    Last Post: October 12th 2009, 10:14 AM
  2. Convergence Proof
    Posted in the Calculus Forum
    Replies: 1
    Last Post: October 20th 2008, 12:16 PM
  3. Proof of convergence
    Posted in the Calculus Forum
    Replies: 4
    Last Post: April 13th 2008, 08:59 PM
  4. Convergence proof
    Posted in the Advanced Math Topics Forum
    Replies: 1
    Last Post: October 31st 2007, 01:17 PM
  5. Proof of Convergence
    Posted in the Calculus Forum
    Replies: 3
    Last Post: December 4th 2006, 10:29 AM

Search Tags


/mathhelpforum @mathhelpforum