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How to calculate this series

  1. Jun 6, 2005 #1
    Is it possible to get a analytical result for this series? It looks simple:

    [itex]\sum_{k=1} ^t a^{t-k}b^{k-1} [/itex]

    Thanks a lot!
     
  2. jcsd
  3. Jun 6, 2005 #2

    dextercioby

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    Hmm.Use [ tex ] & [ /tex ] commands (without the spaces) for opening & closing tex tags.

    That's no series,it's a polynomial in 2 variables.

    Daniel.
     
  4. Jun 6, 2005 #3
    Thanks for your corection. It's my first time to post message here.

    Yes, this is a polynomial, with all coeficient as 1. Is there some method to deal with it?
     
  5. Jun 6, 2005 #4

    dextercioby

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    Yes,try to write some terms in the sum and then see whether you recognize something familiar.

    Daniel.
     
  6. Jun 6, 2005 #5
    Thanks for your hints. I think it can be calculated in this way:
    [tex]\sum_{k=1} ^t a^{t-k}b^{k-1} = \frac{a^t}{b}\sum_{k=1} ^t (\frac{b}{a})^{k}[/tex]
     
  7. Jun 6, 2005 #6

    dextercioby

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    It's easier this way:

    [tex] \sum_{k=1}^{t} a^{t-k}b^{k-1}=a^{t-1}b^{0}+a^{t-2}b^{1}+...+a^{1}b^{t-2}+a^{0}b^{t-1}=\frac{a^{t}-b^{t}}{a-b} [/tex]

    with "t" uneven.

    Daniel.
     
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