Results 1 to 2 of 2

Math Help - sum of the infinite series!

  1. #1
    Newbie
    Joined
    Apr 2008
    Posts
    16

    sum of the infinite series!

    We have to find the value of the sum: -
    [ (n)/(n+1)! ] when n goes from 1 <= n <= infinity...

    I know that converges, and that approaches to some approximate value, but not able to figure out the exact value of it...I think that converges to .99999 or 1.000...please help!

    Thanks!
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Lord of certain Rings
    Isomorphism's Avatar
    Joined
    Dec 2007
    From
    IISc, Bangalore
    Posts
    1,465
    Thanks
    6
    Quote Originally Posted by vikramtiwari View Post
    We have to find the value of the sum: -
    [ (n)/(n+1)! ] when n goes from 1 <= n <= infinity...

    I know that converges, and that approaches to some approximate value, but not able to figure out the exact value of it...I think that converges to .99999 or 1.000...please help!

    Thanks!

    Method 1: (Telescoping)


    <br />
\sum_{1}^{\infty} \frac{n}{(n+1)!} =  \sum_{1}^{\infty} \left(\frac{n+1 - 1}{(n+1)!}\right) = \sum_{1}^{\infty} \left(\frac{1}{n!} - \frac{1}{(n+1)!}\right)

    You should be able to continue from here

    Method 2: (Power series)
    If you know e^x power series,

     \frac{e^x - 1}{x} = \sum_{0}^{\infty} \frac{x^{n}}{(n+1)!}

    Differentiate on both sides and then set x=1.

    The RHS is your series:

    \left(\sum_{0}^{\infty} \frac{x^{n}}{(n+1)!}\right)' \bigg{|}_{x=1}= \sum_{1}^{\infty} \frac{nx^{n-1}}{(n+1)!}\bigg{|}_{x=1} =  \sum_{1}^{\infty} \frac{n}{(n+1)!}

    The LHS is:

     \left(\frac{e^x - 1}{x}\right)'\bigg{|}_{x=1} = \frac{xe^x - (e^x - 1)}{x^2}\bigg{|}_{x=1} = \frac{e - e + 1}1 = 1

    So  \sum_{1}^{\infty} \frac{n}{(n+1)!} = 1 Q.E.D
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. [SOLVED] Fourier series to calculate an infinite series
    Posted in the Differential Geometry Forum
    Replies: 1
    Last Post: August 4th 2010, 01:49 PM
  2. Replies: 2
    Last Post: May 26th 2010, 09:17 AM
  3. Replies: 7
    Last Post: October 12th 2009, 10:10 AM
  4. Replies: 2
    Last Post: September 16th 2009, 07:56 AM
  5. Replies: 1
    Last Post: May 5th 2008, 09:44 PM

Search Tags


/mathhelpforum @mathhelpforum