1. **Problem Statement:** We are given the infinite series $$\sum_{n=1}^{\infty} \frac{(3x - 2)^n}{n}$$ and asked to find its series representation as a function. 2. **Recall the formula:** The series $$\sum_{n=1}^{\infty} \frac{z^n}{n}$$ for $$|z| < 1$$ is known to represent $$-\ln(1 - z)$$. 3. **Apply the formula:** Here, $$z = 3x - 2$$. So, the series can be written as: $$\sum_{n=1}^{\infty} \frac{(3x - 2)^n}{n} = -\ln(1 - (3x - 2)) = -\ln(1 - 3x + 2) = -\ln(3 - 3x)$$ 4. **Simplify the argument:** Factor out 3: $$-\ln(3(1 - x)) = -\ln(3) - \ln(1 - x)$$ 5. **Domain considerations:** The series converges when $$|3x - 2| < 1$$, which means: $$-1 < 3x - 2 < 1$$ $$1 < 3x < 3$$ $$\frac{1}{3} < x < 1$$ 6. **Final series representation:** $$\sum_{n=1}^{\infty} \frac{(3x - 2)^n}{n} = -\ln(3) - \ln(1 - x)$$ for $$x$$ in $$\left(\frac{1}{3}, 1\right)$$. This shows the infinite series sums to a logarithmic function shifted by a constant.