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Find the sum of the following convergent series

  1. Mar 25, 2009 #1

    PAR

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    Sorry about the title, if possible please change it

    1. Find the sum of the following convergent series

    [tex]\sum_{j=0}^{\infty}(-1)^{j}(2/3)^{j}[/tex]




    2. [tex]\sum_{j=0}^{\infty}c^{j} = 1/(1-c) if |c| < 1[/tex]



    3. The attempt at a solution


    [tex]\sum_{j=0}^{\infty}(-1)^{j}(2/3)^{j} = 1 - 2/3 + (2/3)^{2} + ...

    = 1 - (2/3 + (2/3)^{3} + ... ) + ((2/3)^{2} + (2/3)^{4} + ...) [/tex]

    Using the geometric series,
    [tex]\sum_{j=0}^{\infty}(2/3)^{j} = 1 + (2/3) + (2/3)^{2} + ... = 1/(1-(2/3)) = 3
    = 1 + (2/3 + (2/3)^{3} + ...) + ((2/3)^{2} + (2/3)^{4} + ...) [/tex]

    [tex](2/3)^{2} + (2/3)^{4} + ...) = 2 - (2/3 + (2/3)^{3} + ...) [/tex]

    substituting into the original problem:

    [tex]\sum_{j=0}^{\infty}(-1)^{j}(2/3)^{j} = 1 - (2/3 + (2/3)^{3} + ... ) + (2 - (2/3 + (2/3)^{3} + ...)

    = 3 - 2(2/3 + (2/3)^{3} + ... ) [/tex]

    Now i dont know what to do, would like some help, thanks!
     
    Last edited: Mar 25, 2009
  2. jcsd
  3. Mar 25, 2009 #2
    Re: rrr

    Just plug write (-1)^j(2/3)^j as (-2/3)^j and use c=-2/3. c doesn't have to be positive, just of absolute value less than one.
     
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