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Let's see what the generating functions are for some very simple sequences. The simplest of all: 1, 1, 1, 1, 1, …. What does the generating series look like? It is simply \(1 + x + x^2 + x^3 + x^4 + \cdots\text{.}\) Now, can we find a closed formula for this power series? Yes! This particular series is really just a geometric series with common ratio \(x\text{.}\) So if we use our “multiply, shift and subtract” technique from Section 2.2, we have

\begin{align*} S \amp = 1 + x + x^2 + x^3 + \cdots\\ \underline{-\qquad xS} \amp \underline{\,\, = ~~~~~~ x + x^2 + x^3 + x^4 + \cdots}\\ (1-x)S \amp = 1 \end{align*}

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