1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: 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


    User Avatar
    Science Advisor
    Homework Helper

    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


    User Avatar
    Science Advisor
    Homework Helper

    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
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook