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12-17-2013
07:32 AM

12-17-2013
07:32 AM

Closed form possible?

Hi all,

I'm attempting to derive an expression for an infinite sum of a series of 2nd order polynomials (see attachment). The independent variable "s" is scaled for each successive term of the sum. The problem is a filter bank with an infinite number of individual overlapping filters with each scaled in frequency.

I know Mathcad can crunch the expression to give me a numerical answer for a value of s using symbolic evaluation "->" but it won't give me a numerical evaluation using "=". Also, it doesn't seem to be able to provide a closed form of the infinite sum (which is what I would like if it were possible).

You will see that I have attempted to break the expression down using a partial fraction expansion so the infinite sum boils down to separate sums of 1st order polynomials but this hasn't really helped.

I lack the know how to figure out if this problem is actually solvable.

Thanks in advance!

Dave

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12-17-2013
09:23 AM

12-17-2013
09:23 AM

Re: Closed form possible?

Without addressing a closed for solution,

If you define a finction with a finite number for the summation (instead of infinity) you can get a numeric answer. For the values in your sheet this answer collapses to the symbolic evaluation at about 10,000 terms. Don't know if that helps you or not. Looks like at about 2000 terms the imaginary part of the summation disappears.

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12-17-2013
09:49 AM

12-17-2013
09:49 AM

Re: Closed form possible?

Hi Fred,

thanks for that. Yes it is interesting and helpful to see how many terms are really necessary before things start to "converge" rather than concentrate on trying to get a closed solution (which would be nice if such a thing exists).

Regards,

Dave

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