Results 1 to 3 of 3
Like Tree2Thanks
  • 1 Post By Lockdown
  • 1 Post By Soroban

Math Help - Sum of a given series

  1. #1
    Member
    Joined
    Feb 2011
    Posts
    83
    Thanks
    2

    Sum of a given series

    Hello

    I was wondering if there was any way to calculate the sum of the series

    \sum_{k=1}^\infty \frac{k}{2^k}

    I know it is convergent by the ratio test, but it there anyway to finding out what sum it converges to?

    This series came up when I was calculating the expected value of a random variable.

    (Wolfram Alpha says it is 2, which makes sense in context of the problem)
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Newbie
    Joined
    Feb 2013
    From
    Home
    Posts
    18
    Thanks
    2

    Re: Sum of a given series

    You can prove that the sum converges to 2 by considering the partial sums.
    It turns out that (I encourage you to prove this):
    \sum_{k=1}^n \frac{k}{2^k} = \frac{2^{n+1}-n-2}{2^n}.

    Hence, taking the limit as n tends to infinity, you can see that the sum converges to 2.
    Thanks from Jame
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Super Member

    Joined
    May 2006
    From
    Lexington, MA (USA)
    Posts
    11,735
    Thanks
    642

    Re: Sum of a given series

    Hello, Jame!

    I was wondering if there was any way to calculate the sum of the series

    . . \sum_{k=1}^\infty \frac{k}{2^k}

    \begin{array}{ccccc}\text{We have: }& S & =& \frac{1}{2} + \frac{2}{2^2} + \frac{3}{2^3} + \frac{4}{2^4} + \cdots \\ \\[-4mm]   \text{Multiply by }\frac{1}{2}\!: & \frac{1}{2}S &=& \quad\;\; \frac{1}{2^2} + \frac{2}{2^3} + \frac{3}{2^4} + \cdots \\ \\[-4mm] \text{Subtract: }& \frac{1}{2}S &=& \frac{1}{2} + \frac{1}{2^2} + \frac{1}{2^3} + \frac{1}{2^4} + \cdots \end{array}

    The right side is a geometric series
    . . with first term a = \tfrac{1}{2} and common ratio r = \tfrac{1}{2}.
    Its sum is: . \frac{\frac{1}{2}}{1-\frac{1}{2}} \:=\:1

    Therefore: . \tfrac{1}{2}S \:=\:1 \quad\Rightarrow\quad S \:=\:2

    Thanks from Jame
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 7
    Last Post: December 9th 2012, 07:12 PM
  2. Replies: 2
    Last Post: May 22nd 2012, 05:57 AM
  3. Replies: 3
    Last Post: September 29th 2010, 06:11 AM
  4. Replies: 0
    Last Post: January 26th 2010, 08:06 AM
  5. Replies: 1
    Last Post: May 5th 2008, 09:44 PM

Search Tags


/mathhelpforum @mathhelpforum