# Is there a closed form of this expression?

by blahdeblah
Tags: expression, form
 P: 9 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. $$H(s)=\sum_{n=-\infty}^\infty \frac{k^n}{k^n+a/s}$$ Thanks, D
 Emeritus Sci Advisor PF Gold P: 4,500 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
 P: 9 Sorry, the process of posting this, made me think of something which might be helpful: $$H(s)=\sum_{n=-\infty}^\infty \frac{1}{1+a/(s.k^n)}$$ so... $$H(s)=\sum_{n=-\infty}^\infty \frac{1}{1+a.k^{-n}/s}$$ I might be able to google this one as it looks a bit more like a standard form of something.
P: 9
Is there a closed form of this expression?

 Quote by Office_Shredder 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
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.
 P: 9 Here is my original problem: $$FB(s)=\sum_{n=-\infty}^\infty \frac{a(k^ns)}{(k^ns)^2+a(k^ns)+1}$$ This is what I really want to obtain the closed form solution for.
P: 1,666
 Quote by blahdeblah Here is my original problem: $$FB(s)=\sum_{n=-\infty}^\infty \frac{a(k^ns)}{(k^ns)^2+a(k^ns)+1}$$ This is what I really want to obtain the closed form solution for.
Entirely inadequate. Please restate the question precisely defining what a, k, and s are and not just "numbers" either.
 P: 9 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.

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