Is there a closed form of this expression?

In summary, the conversation discusses the possibility of finding a closed form expression for the infinite sum given by H(s)=\sum_{n=-\infty}^\infty \frac{a(k^ns)}{(k^ns)^2+a(k^ns)+1}, where a, k, and s are real and imaginary scalars. The conversation also mentions the possibility of breaking down the original expression into partial fraction sums to obtain a closed form solution. Some additional clarification on the variables and the original problem is also requested.
  • #1
blahdeblah
9
0
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
 
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  • #2
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
 
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  • #3
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.
 
  • #4
Office_Shredder said:
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.
 
Last edited:
  • #5
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.
 
  • #6
blahdeblah said:
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.

Entirely inadequate. Please restate the question precisely defining what a, k, and s are and not just "numbers" either.
 
  • #7
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:

1. What is a closed form expression?

A closed form expression is a mathematical formula or expression that can be written using a finite number of standard mathematical operations, such as addition, subtraction, multiplication, division, and exponentiation, without the use of limits or infinite series.

2. Why is it important to find a closed form expression?

Finding a closed form expression can provide a more efficient and concise way of representing and solving a mathematical problem. It allows for easier manipulation and analysis of the expression, making it a valuable tool in many areas of mathematics and science.

3. Is it always possible to find a closed form expression?

No, not all mathematical expressions have a closed form. Some expressions may require the use of infinite series or limits, making them impossible to write in a finite form. However, in many cases, it is possible to approximate a solution using numerical methods.

4. How do you determine if a mathematical expression has a closed form?

There is no simple method for determining if a mathematical expression has a closed form. It often requires a combination of mathematical techniques and intuition. Some common methods include looking for patterns, simplifying the expression, and using known identities and theorems.

5. Can a closed form expression provide an exact solution?

Yes, a closed form expression can provide an exact solution to a mathematical problem. It is a finite, precise representation of the problem that can be evaluated to obtain the exact solution. However, in some cases, the closed form expression may be too complex or impractical to use, and an approximate solution may be preferred.

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