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I am relatively new to all of this so I apologize if I ask this question poorly...

I have figured out a nice closed solution form for series in the form of $P(n) / a^n$ where $p(n)$ is a polynomial and $a$ is a number greater than 1. Similarly I found a nice solution form for ones that also have the alternating component of $(-1)^n$.

Normally, when we are taught in school, you only hear about finding exact values for geometric and telescoping series, assuming they converge. But my solutions do not fall into that camp, at least I don't think they do, so how do I go about finding in the literature what I am doing, like where do I look to see if my findings already exist? Like do I just submit somewhere for publication and find out that way?

Hopefully for added clarity, what I am talking about is that i found a simple 'formula' for finding the exact value to something like this (forgive me, I don't understand math text):

$\sum_n ((-1)^n(3n^4+12n^2+6n-8)/(4^n))$ where $n$ goes from 1 to infinity.

A rural reader
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    Does this answer your question? Summation of series $\sum_{n=1}^\infty \frac{n^a}{b^n}$? Note that you can derive closed form expressions not only for the limit, but even for the partial sums. – dxiv Apr 28 '23 at 20:42
  • I think so. Polylograrothm. I'll need to read up on this. The solution representation is different than mine but this looks much more compact (clean?) Than mine. Thank you for sharing. –  Apr 28 '23 at 20:48
  • There are several ways besides polylog to derive and use the result, and the answers under the other question demonstrate a few of them. – dxiv Apr 28 '23 at 20:58
  • https://math.stackexchange.com/questions/1743544/are-there-some-techniques-which-can-be-used-to-show-that-a-sum-does-not-have-a/2329022#2329022 – IV_ Apr 29 '23 at 09:04

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This is a linear recurrence, a sequence with a rational generating function. Its sum is also a linear recurrence. You can determine the characteristic polynomials for the two and then come up with a closed-form formula which will be the sum of a finite number of terms, each of which is a polynomial times an exponential. The base of the exponential and the coefficients of the polynomial will be algebraic numbers if your original numbers are.

There are texts on recurrences or you could read a combinatorics textbook (maybe generatingfunctionology).

Charles
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