The Cauchy Integral Test (or Integral Test) states that if $f(n)$ is a positive, continuous, and decreasing function for all sufficiently large $n$, then the convergence of the improper integral
$\int_{a}^{\infty} f(x) \, dx$ determines the convergence of the series
$\sum_{n=a}^{\infty} f(n).$

Step 1: Define the Function

We define: $f(n) = \frac{\ln(n^2)}{n}.$ Since $\ln(n^2) = 2\ln(n)$, we rewrite: $f(n) = \frac{2\ln(n)}{n}.$ This function is positive for $n \geq 2$ and is continuous and decreasing for sufficiently large $n$.

Step 2: Evaluate the Improper Integral

We evaluate: $I = \int_{2}^{\infty} \frac{2\ln(x)}{x} dx.$ Using substitution, let: $u = \ln(x), \quad \text{so that} \quad du = \frac{dx}{x}.$ Then the integral transforms into: $I = \int_{\ln(2)}^{\infty} 2u \, du.$ Evaluating: $\int 2u \, du = u^2.$ Applying limits: $I = \left[ u^2 \right]_{\ln(2)}^{\infty} = \infty^2 - (\ln(2))^2 = \infty.$ Since the integral diverges, the original series: $\sum_{n=2}^{\infty} \frac{\ln(n^2)}{n}$ also diverges by the Integral Test.

---

Conclusion:

By the Cauchy Integral Test, the given series diverges.

Would you like a more detailed explanation or another example?

---

Related Questions:

1. How do you apply the Cauchy Integral Test to other logarithmic series?
2. What are the conditions for the Integral Test to be valid?
3. Can the p-series test be used for a comparison with this series?
4. How does the Rate of Divergence of this series compare with the harmonic series?
5. What happens if we modify the numerator to $\ln(n)$ instead of $\ln(n^2)$?

Tip: When applying the Integral Test, always check that the function is positive, continuous, and decreasing!