I need to find the coefficient of $x^5$ in $\frac{2x}{1 - 3x}.$ I have found two different solutions, but I am not sure which is correct. Is the coefficient of $x^5$ given by $3^5 \cdot 2x$ under the interpretation $2x \cdot \frac{1}{1 - 3x},$ or is it given by $2 \cdot 3^4$ under the interpretation $2 \frac{x}{1-3x}?$
2026-04-02 05:18:01.1775107081
Finding Coefficients via Generating Functions
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Consider the formal power series $f(x) = \frac 1 {1 - x} = \sum_{n = 0}^\infty x^n.$ We want to find the coefficient of $x^5$ in $g(x) = \frac{2x}{1 - 3x},$ so we can use the formal power series of $f(x)$ to find the formal power series of $g(x).$ Explicitly, we have that $g(x) = 2x f(3x).$ Can you finish the solution from here?