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Homework Help: Finding the sum of an infinite series

  1. Dec 4, 2009 #1
    Find the sum of the infinite series

    [tex]\sum _{n=1}^{\infty } \left( i /2\right) ^{2\,n}[/tex]

    I just can't seem to get started on this problem, so I was hoping somebody could give me a hint, as to what methods i should read up on.
  2. jcsd
  3. Dec 4, 2009 #2
    A good place to start would be to consider what (i/2)^2 is, that simplifies the sum considerably. You also need the formula for geometric series, that you can find for example from wikipedia.
  4. Dec 5, 2009 #3
    Actually, break i into its polar form which is [tex] i^2^n = e^i^n^(^p^i^) [/tex] . Now expand using Euler's identity and you are left only with [tex] cos\ n(pi) = (-1)^n [/tex] From there it is a geometric series.
  5. Dec 5, 2009 #4


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    Science Advisor

    Actually, there is no need to worry about "i". [itex](i/2)^{2n}= (i^2/4)^n= (-1/4)^n[/itex] so this is a purely real geometric series.
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