1. **Stating the problem:** We want to understand Cauchy's nth root test, which is a method to determine the convergence or divergence of an infinite series $\sum a_n$. 2. **Formula:** The test uses the limit $$L = \lim_{n \to \infty} \sqrt[n]{|a_n|}$$ 3. **Rules:** - If $L < 1$, the series converges absolutely. - If $L > 1$, the series diverges. - If $L = 1$, the test is inconclusive. 4. **Explanation:** The nth root test examines the behavior of the terms $a_n$ raised to the power $1/n$. If these terms shrink fast enough (i.e., $L < 1$), the series converges. If they do not shrink or grow (i.e., $L \geq 1$), the series diverges or the test is inconclusive. 5. **Example:** Consider the series $\sum \left(\frac{1}{2}\right)^n$. - Compute $L = \lim_{n \to \infty} \sqrt[n]{\left|\left(\frac{1}{2}\right)^n\right|} = \lim_{n \to \infty} \left(\frac{1}{2}\right) = \frac{1}{2}$. - Since $L = \frac{1}{2} < 1$, the series converges. This test is especially useful for series with terms involving powers or factorials.