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Feeling like a tool for not remembering this series.

  1. Oct 13, 2012 #1
    So I'm trying to find the value of this finite series and I'm feeling like a tool for not being able to remember the way to calculate the result...


    S = 1/(1+x) + 2/(1+x)2 + 3/(1+x)3 +........+ 10/(1+x)10

    I know it's a geometric series, but I cannot figure out what to use as my initial term.

    Thanks.
     
  2. jcsd
  3. Oct 13, 2012 #2

    tiny-tim

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    hi trap101! :smile:

    no, it's not geometric (because of the 1 2 3 …)

    try differentiating or integrating it :wink:
     
  4. Oct 13, 2012 #3

    SammyS

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    Yes. Integrate the general term of the series, [itex]\displaystyle \frac{k}{(1+x)^k}=
    k(1+x)^{-k}\ .[/itex]
     
  5. Oct 13, 2012 #4


    Well that's not good because it was actually just a portion of a larger problem I was working on. Observe #5 in the attachment:


    So essentially I had done some manipualtions to the point that I obtained:

    C/PV0 [ the series from above] + F/(1+x)10

    where C/PV, F are constants.

    I followed your suggestion of integrating the general term and got:

    k(1+x)1-k/(1-k).........but how does this help in obtaining a sum for that series?
     
  6. Oct 13, 2012 #5
    sorry here was the attachment..
     

    Attached Files:

    • hw1.pdf
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  7. Oct 13, 2012 #6
    sorry here was the attachment..
     
  8. Oct 14, 2012 #7

    tiny-tim

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    hi trap101! :smile:
    it doesn't, the k/(1-k) messes it up!

    you need to fiddle about with the original series to get something that integrates to a straightforward factor-less ∑ (1+x)k :wink:
     
  9. Oct 14, 2012 #8
    No , tiny-tim !! No need to differentiate or integrate it !

    trap101 , the series is not geometric or arithmetic , but it is "Arithmetico-Geometric series" , and there is a way of solving it.

    S = 1/(1+x) + 2/(1+x)2 + 3/(1+x)3 +........+ 10/(1+x)10 ..... equation 1

    Multiply both the sides of the equation by the common ratio , 1/(1+x) and get another equation.

    Arithmetico-Geometric series are of form :

    a, (a+d)r , (a+2d)r2 , ......... , {a+(n-1)d} rn-1
     
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