1. **Problem statement:** Determine whether the improper integral $$\int_1^{\infty} \frac{\ln x}{x} \, dx$$ converges or diverges. 2. **Recall the integral test and behavior of functions:** For improper integrals of the form $$\int_a^{\infty} f(x) \, dx$$, we analyze the limit $$\lim_{t \to \infty} \int_a^t f(x) \, dx$$. 3. **Set up the integral:** $$\int_1^{\infty} \frac{\ln x}{x} \, dx = \lim_{t \to \infty} \int_1^t \frac{\ln x}{x} \, dx$$ 4. **Use substitution to find the antiderivative:** Let $$u = \ln x$$, then $$du = \frac{1}{x} dx$$. 5. **Rewrite the integral:** $$\int \frac{\ln x}{x} dx = \int u \, du = \frac{u^2}{2} + C = \frac{(\ln x)^2}{2} + C$$ 6. **Evaluate the definite integral:** $$\int_1^t \frac{\ln x}{x} dx = \left[ \frac{(\ln x)^2}{2} \right]_1^t = \frac{(\ln t)^2}{2} - \frac{(\ln 1)^2}{2} = \frac{(\ln t)^2}{2} - 0$$ 7. **Analyze the limit as $$t \to \infty$$:** $$\lim_{t \to \infty} \frac{(\ln t)^2}{2} = \infty$$ 8. **Conclusion:** Since the limit diverges to infinity, the improper integral $$\int_1^{\infty} \frac{\ln x}{x} \, dx$$ diverges. **Final answer:** The integral diverges.