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Series convergence representation

  1. Nov 15, 2007 #1
    1. The problem statement, all variables and given/known data
    [tex]\sum_{n=0}^\infty (0.5)^n * e^{-jn}[/tex]

    converges into


    Prove the convergence.

    2. Relevant equations

    Power series, and perhaps taylor & Macclaurin representation of series.

    3. The attempt at a solution

    This isn't a homework problem, actually. I just saw this series on the poster and wondered why this is the case (I haven't done series for almost 2 years).

    I know for sure that the series has to converge since the [tex]0.5^n[/tex] term approaches 0 as n goes to infinity, but I don't understand how the series written above converges into [tex]\frac{1}{1-0.5e^{-jn}}[/tex]. Can anyone explain?
  2. jcsd
  3. Nov 15, 2007 #2


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    There shouldn't be an n in your final answer, obviously.
    But isn't this just an ordinary geometric series?
    [tex]\sum_{n = 0}^\infty x^n = \frac{1}{1 - x}[/tex]
  4. Nov 15, 2007 #3
    That's what I was thinking, except that the series is multiplied by an exponential term (with n). And sorry, there was a mistake - there shouldn't be n in the final answer.
  5. Nov 15, 2007 #4


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    Don't get confused over a rewriting of something you already knew :smile:
    If I'd write it as
    [tex]\sum_{n = 0}^\infty \left( \tfrac12 e^{-j} \right)^n, [/tex]
    which is obviously possible since [itex](e^a)^b = e^{ab}[/itex], would you see it's the same?
  6. Nov 15, 2007 #5
    [Hits Head]

    Doh, of course. Thanks
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