Question

Use the Root Test to determine the convergence or divergence of the series. $$ \sum_{n=1}^{\infty}\left(\frac{\ln n}{n}\right)^{n} $$

Short answer

The series converges.

Step-by-step solution

Apply the Root Test

The Root Test can be applied to the given series. This involves taking the nth root of the absolute value of each term: \[ \lim_{n \to \infty} \sqrt[n]{|a_n|} = \lim_{n \to \infty} \sqrt[n]{\left| \left(\frac{\ln n}{n}\right)^{n} \right|} = \lim_{n \to \infty} \frac{\ln n}{n} \]

Evaluate the Limit

Next, we need to evaluate this limit as n approaches infinity. However, this is an indeterminate form of type 0/0, so we must use L'Hopital's rule. In order to do so, we need to take the derivative of numerator and denominator. Using L'Hopital's rule, the limit is equivalent to: \[ \lim_{n \to \infty} \frac{1/n}{1} = \lim_{n \to \infty} \frac{1}{n} \]

Apply the Rules of the Root Test

Finally, apply the rules of the Root Test. The Root Test states that if \(\lim_{n \to \infty} \sqrt[n]{|a_n|}>1\), the series diverges. If \(\lim_{n \to \infty} \sqrt[n]{|a_n|}<1\), the series converges. If the limit equals 1, the test is inconclusive. In this case, as n approaches infinity, the result from Step 2 approaches 0, which is less than 1, hence the series converges according to the root test.