Cauchy’s Root Test

Cauchy's Root Test is a convergence test used to determine whether an infinite series converges or diverges. The test is especially useful when dealing with series that involve powers of $n$, such as:

$\sum_{n=1}^{\infty} a_n$

where $a_n$ is the $n$-th term of the series. The test involves analyzing the $n$-th root of the absolute value of the terms in the series.

Statement of Cauchy's Root Test:

Given a series $\sum_{n=1}^{\infty} a_n$, define:

$L = \lim_{n \to \infty} \sqrt[n]{|a_n|}$

The test then states the following:

1. If $L < 1$: The series converges absolutely.
2. If $L > 1$: The series diverges.
3. If $L = 1$: The test is inconclusive, and we need to use other methods to determine convergence or divergence.

Detailed Steps for Applying Cauchy’s Root Test:

1. Find the general form of the series $a_n$.
2. Compute the $n$-th root of the absolute value of $a_n$, i.e., $\sqrt[n]{|a_n|}$.
3. Take the limit of this $n$-th root as $n \to \infty$, and find the value of $L$.
4. Interpret the result based on the conditions mentioned above.

Example:

Let’s consider the series:

$\sum_{n=1}^{\infty} \frac{n^n}{3^n}$

We need to determine whether this series converges or diverges using Cauchy's Root Test.

Step 1: Identify $a_n$

Here, the general term of the series is:

$a_n = \frac{n^n}{3^n}$

Step 2: Apply the $n$-th root

We need to compute:

[ \sqrt[n]{|a_n|} = \sqrt[n]{\left| \frac{n^n}{3^n} \right|} ]

This simplifies as:

$\sqrt[n]{\frac{n^n}{3^n}} = \frac{n}{3}$

Step 3: Take the limit as $n \to \infty$

Now, we calculate the limit:

$L = \lim_{n \to \infty} \frac{n}{3}$

As $n \to \infty$, we see that $L = \infty$.

Step 4: Interpret the result

Since $L = \infty$ and $L > 1$, the series diverges by Cauchy’s Root Test.

Conclusion:

The series $\sum_{n=1}^{\infty} \frac{n^n}{3^n}$ diverges because the limit $L$ is greater than 1.

---

Questions to Explore Further:

1. How does Cauchy’s Root Test compare to the Ratio Test?
2. Can we apply Cauchy’s Root Test to series with terms involving factorials?
3. How do we handle the case when $L = 1$?
4. What are other convergence tests that can be used when the Root Test is inconclusive?
5. How can we modify the series $\sum \frac{n^n}{3^n}$ to make it converge?

Tip:

When dealing with series involving powers, exponentials, or factorials, Cauchy’s Root Test is particularly useful. However, if the limit $L = 1$, try combining it with other tests like the Ratio Test or Integral Test to gain further insight into the behavior of the series.