Is there a closed form of this expression?

  1. Hi,
    (not homework/academic)
    Is a closed form of the following expression possible? Either way, some pointers in the right direction would be really helpful.

    [tex]
    H(s)=\sum_{n=-\infty}^\infty \frac{k^n}{k^n+a/s}
    [/tex]

    Thanks,
    D
     
  2. jcsd
  3. Office_Shredder

    Office_Shredder 4,498
    Staff Emeritus
    Science Advisor
    Gold Member

    Are a, s and k numbers? If so then this sum is divergent - as n goes to +/- infinity, the summand converges to 1 depending on whether k is larger than or smaller than 1
     
    1 person likes this.
  4. Sorry, the process of posting this, made me think of something which might be helpful:
    [tex]
    H(s)=\sum_{n=-\infty}^\infty \frac{1}{1+a/(s.k^n)}
    [/tex]
    so...
    [tex]
    H(s)=\sum_{n=-\infty}^\infty \frac{1}{1+a.k^{-n}/s}
    [/tex]
    I might be able to google this one as it looks a bit more like a standard form of something.
     
  5. a is a constant, yes and s is a variable (actually frequency in my application).

    I already know from experimentation in Mathcad that an expression using this basic block produces a reasonable result (I suppose I should say bounded). The original expression is 2nd order and the associated response in s tends to 0 as s->0 and as s->INF. I managed to break down the original into a partial fraction sum so could treat it as 2 independent infinite sums of 1st order functions like the one shown. I didn't consider if/whether the 1st order expressions would diverge or not.

    Perhaps I should post my original problem.
     
    Last edited: Dec 17, 2013
  6. Here is my original problem:
    [tex]
    FB(s)=\sum_{n=-\infty}^\infty \frac{a(k^ns)}{(k^ns)^2+a(k^ns)+1}
    [/tex]
    This is what I really want to obtain the closed form solution for.
     
  7. Entirely inadequate. Please restate the question precisely defining what a, k, and s are and not just "numbers" either.
     
  8. k is a real scalar > 1
    a is a real scalar > 0
    s is a imaginary scalar > 0

    My first post asks if there is a closed form expression of the infinite sum given. If the answer is yes, then some guidance in the right direction to help to obtain it would be very helpful.

    If a solution is indeed available then I think it follows that the expression in my last post (#6) can be solved.
     
    Last edited: Dec 18, 2013
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