Results 1 to 7 of 7

Math Help - Infinite series : convergence.

  1. #1
    MHF Contributor arbolis's Avatar
    Joined
    Apr 2008
    From
    Teyateyaneng
    Posts
    1,000
    Awards
    1

    Infinite series : convergence.

    \sum_{n=1}^{+\infty} \frac{2^nn!}{n^n}. I must show whether it converges or not. Using my intuition I think it converges. I tried to use the root test, the only test I think that works.
    So I'm stuck when I must find the limit when n tends to +\infty of \frac{2\cdot n!^{\frac{1}{n}}}{n}. Does the limit of n!^\frac{1}{n}=\frac{1}{e}, when n tends to +\infty? If yes, how to prove it using maths of calculus II?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor Mathstud28's Avatar
    Joined
    Mar 2008
    From
    Pennsylvania
    Posts
    3,641
    Quote Originally Posted by arbolis View Post
    \sum_{n=1}^{+\infty} \frac{2^nn!}{n^n}. I must show whether it converges or not. Using my intuition I think it converges. I tried to use the root test, the only test I think that works.
    So I'm stuck when I must find the limit when n tends to +\infty of \frac{2\cdot n!^{\frac{1}{n}}}{n}. Does the limit of n!^\frac{1}{n}=\frac{1}{e}, when n tends to +\infty? If yes, how to prove it using maths of calculus II?
    Root test.

    And just consider that all this is limit is is

    \lim_{n\to\infty}(2^n)^{\frac{1}{n}}\cdot\lim_{n\t  o\infty}\left(\frac{n!}{n^n}\right)^{\frac{1}{n}}=  2\lim_{n\to\infty}\left(\frac{n!}{n^n}\right)^{\fr  ac{1}{n}}

    The last limit we have gone over.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    MHF Contributor

    Joined
    Aug 2006
    Posts
    18,642
    Thanks
    1594
    Awards
    1
    \left( {\sqrt[n]{{\frac{{2^n n!}}{{n^n }}}}} \right) \to \frac{2}{e}
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor Mathstud28's Avatar
    Joined
    Mar 2008
    From
    Pennsylvania
    Posts
    3,641
    Quote Originally Posted by arbolis View Post
    \sum_{n=1}^{+\infty} \frac{2^nn!}{n^n}. I must show whether it converges or not. Using my intuition I think it converges. I tried to use the root test, the only test I think that works.
    So I'm stuck when I must find the limit when n tends to +\infty of \frac{2\cdot n!^{\frac{1}{n}}}{n}. Does the limit of n!^\frac{1}{n}=\frac{1}{e}, when n tends to +\infty? If yes, how to prove it using maths of calculus II?
    you could just say that \frac{2^nn!}{n^n}\sim\left(\frac{2}{e}\right)^n

    By stirlings approximation
    Follow Math Help Forum on Facebook and Google+

  5. #5
    is up to his old tricks again! Jhevon's Avatar
    Joined
    Feb 2007
    From
    New York, USA
    Posts
    11,663
    Thanks
    3
    the ratio test also works nicely here. you end up with the limit being \frac 2e as well, which is less than 1, of course. it is easier to see the limit using this way, in my humble opinion
    Follow Math Help Forum on Facebook and Google+

  6. #6
    MHF Contributor arbolis's Avatar
    Joined
    Apr 2008
    From
    Teyateyaneng
    Posts
    1,000
    Awards
    1
    Thanks to all!! This means that \sum_{n=1}^{+\infty} \frac{3^nn!}{n^n} diverges. (I changed the 2 for a 3). As I have to determine it as well, it's already done!
    Follow Math Help Forum on Facebook and Google+

  7. #7
    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 arbolis View Post
    Thanks to all!! This means that \sum_{n=1}^{+\infty} \frac{3^nn!}{n^n} diverges. (I changed the 2 for a 3). As I have to determine it as well, it's already done!
    yes
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Convergence of Infinite Series
    Posted in the Differential Geometry Forum
    Replies: 1
    Last Post: March 6th 2011, 02:46 AM
  2. Convergence of Infinite Series
    Posted in the Calculus Forum
    Replies: 4
    Last Post: June 8th 2010, 03:09 PM
  3. Replies: 7
    Last Post: October 12th 2009, 10:10 AM
  4. convergence of an infinite series
    Posted in the Calculus Forum
    Replies: 1
    Last Post: November 17th 2008, 04:54 AM
  5. Convergence of infinite series
    Posted in the Calculus Forum
    Replies: 2
    Last Post: April 12th 2008, 02:11 PM

Search Tags


/mathhelpforum @mathhelpforum