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A "polynomial geometric" series is one in which the kth term is the product of an mdegree polynomial in k times x^{k}.
_{n} 
(a_{0} + a_{1}k + a_{2}k^{2} + ... + a_{m1}k^{m1} + a_{m}k^{m}) x^{k} 
The way to find such a sum is to telescope the mdegree polynomial by taking successive differences. That is, by multiplying the series by (1x), the resulting series is a polynomial geometric series of degree m1. The first and last terms, don't cancel, however. Then, repeating the process, i.e. multiplying again by (1x), the series eventually becomes purely geometric, with m noncanceling terms at the beginning, and another m noncanceling terms at the end. Then, multiply by (1x) one more time, as you normally do with a geometric series, to make "most" of the terms cancel completely.
It turns out that the noncanceling terms become quite a challenge, especially if n < 2m  that is, when the noncanceling terms overlap. For this reason, I will confine the rest of this article to the case when n approaches infinity, so the noncanceling terms will occur only at the beginning of the series. This doesn't actually cause any loss of generality, though, because a finite series can be expressed as the difference of two infinite series with different starting points.
Let S be the sum of an infinite series of this form,
S = 
_{∞} 
(a_{0} + a_{1}k + a_{2}k^{2} + ... + a_{m1}k^{m1} + a_{m}k^{m}) x^{k} 
Now, multiply the sequence by (1x)^{m+1}, giving
S(1x)^{m+1} =
C(m+1,0) _{∞}
∑
^{k=0}(a_{0} + a_{1}k + a_{2}k^{2} + ... + a_{m1}k^{m1} + a_{m}k^{m}) x^{k}
 C(m+1,1) _{∞}
∑
^{k=1}(a_{0} + a_{1}(k1) + a_{2}(k1)^{2} + ... + a_{m1}(k1)^{m1} + a_{m}(k1)^{m}) x^{k}
. . .
+ (1)^{i}C(m+1,i) _{∞}
∑
^{k=i}(a_{0} + a_{1}(ki) + a_{2}(ki)^{2} + ... + a_{m1}(ki)^{m1} + a_{m}(ki)^{m}) x^{k}
. . .
+ (1)^{m+1}C(m+1,m+1) _{∞}
∑
^{k=m}(a_{0} + a_{1}(km) + a_{2}(km)^{2} + ... + a_{m1}(km)^{m1} + a_{m}(km)^{m}) x^{k}
The beauty of this telescoping method is that all the terms of S(1x)^{m+1} except for the first m+1 terms cancel completely. The only terms left are:
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The webmaster and author of this Math Help site is Graeme McRae.