Let f(x)=∑n=0∞xn2nf(x) = \sum_{n=0}^{\infty} \frac{x^n}{2^n}f(x)=∑n=0∞2nxn and g(x)=∑n=0∞(−1)nxn3ng(x) = \sum_{n=0}^{\infty} \frac{(-1)^n x^n}{3^n}g(x)=∑n=0∞3n(−1)nxn. Find the coefficient of xxx in the product [f(x)]2−[g(x)]2[f(x)]^2 - [g(x)]^2[f(x)]2−[g(x)]2.
12+19\frac{1}{2} + \frac{1}{9}21+91
2⋅12−2⋅(−13)=1+23=532 \cdot \frac{1}{2} - 2 \cdot \left(-\frac{1}{3}\right) = 1 + \frac{2}{3} = \frac{5}{3}2⋅21−2⋅(−31)=1+32=35
12\frac{1}{2}21
23\frac{2}{3}32