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The point is, if you need to find a generating function for the sum of the first \(n\) terms of a particular sequence, and you know the generating function for that sequence, you can multiply it by \(\frac{1}{1-x}\text{.}\) To go back from the sequence of partial sums to the original sequence, you look at the sequence of differences. When you get the sequence of differences you end up multiplying by \(1-x\text{,}\) or equivalently, dividing by \(\frac{1}{1-x}\text{.}\) Multiplying by \(\frac{1}{1-x}\) gives partial sums, dividing by \(\frac{1}{1-x}\) gives differences.