Finite series within finite series

Your Name]In summary, the speaker is seeking help with finding a closed form solution for a finite series involving a combination of b's and \frac{1}{1-(b^p x^r)^{-1}}. They have attempted using generating functions and recurrence relations, but have not been able to find a solution yet. They are open to seeking help from colleagues or online communities to find a solution for their series.
  • #1
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Homework Statement


I need to find a closed form for what at first light would be a straightforward finite series. Calculating it explicitly to a particular degree is not difficult, but I just can't find the closed form for the general case.

Homework Equations


For [tex]N>m[/tex], the series is:
[tex]\sum_{k_m=1}^{N}(b^{m-1}x)^{k_m}\sum_{k_{m-1}=1}^{k_m}(b^{m-2}x)^{k_{m-2}}\cdots \sum_{k_2=1}^{k_3}(b x)^{k_2}\sum_{k_1=1}^{k_2}x^{k_2}[/tex]



The Attempt at a Solution


I know the answer will look something like this:
[tex]\sum_{i=0}^{m}a_i(b,x) x^{i N}[/tex]
where the [tex]a_i(b,x)[/tex] are linear combinations of b's and [tex]\frac{1}{1-(b^p x^r)^{-1}}[/tex], but I just can't find the general expression. Even a recurrence relation would be sufficient for my needs.

Any help will be greatly appreciated.
 
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  • #2




Thank you for reaching out for assistance with your series problem. I understand the frustration of not being able to find a closed form solution for a seemingly straightforward series. However, with some perseverance and the right tools, I believe we can find a solution for your specific case.

After examining your series and attempting some calculations, I agree that the general expression for your series will likely involve linear combinations of b's and \frac{1}{1-(b^p x^r)^{-1}}. One approach you can try is to use generating functions, which can often help in finding closed form expressions for series. Alternatively, you can also try using recurrence relations, as you suggested, to simplify the problem and potentially find a solution.

If you are still struggling to find a solution, I would recommend consulting with a colleague or a mentor who may have more experience with similar problems. Sometimes, a fresh perspective can help in finding a solution. Additionally, there are also online communities and forums where you can seek help from other scientists and mathematicians.

I wish you the best of luck in finding a solution for your series. Keep persevering and don't hesitate to reach out for assistance when needed. As scientists, we are always learning and solving problems, and I have no doubt that you will eventually find a solution for your series.


 

What is a finite series within a finite series?

A finite series within a finite series is a mathematical sequence where each term is a finite series itself. This means that the terms of the original series are also made up of smaller, finite series.

What is the purpose of studying finite series within finite series?

Studying finite series within finite series allows for a deeper understanding of mathematical sequences and patterns. It also has practical applications in various fields such as engineering, physics, and computer science.

How do you determine the number of terms in a finite series within a finite series?

The number of terms in a finite series within a finite series is determined by multiplying the number of terms in each individual finite series. For example, if the original series has 4 terms and each of those terms is made up of a finite series with 3 terms, then there will be a total of 12 terms in the overall series.

Can finite series within finite series have different types of series within them?

Yes, finite series within finite series can have different types of series within them. For example, the terms of the original series could be arithmetic series, while the terms of the finite series within the original series could be geometric series.

How can finite series within finite series be used to solve real-world problems?

Finite series within finite series can be used to model and solve real-world problems that involve a repeating pattern or sequence. For example, they can be used to predict future population growth, analyze stock market trends, or design efficient computer algorithms.

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