Question

Determine the convergence of the given series using the Root Test. If the Root Test is inconclusive, state so and determine convergence with another test. $$\sum_{n=1}^{\infty} \frac{1}{(\ln n)^{n}}$$

Short answer

The series converges by the Root Test.

Step-by-step solution

Identify the General Term

The given series is \( \sum_{n=1}^{\infty} \frac{1}{(\ln n)^{n}} \). Identify the general term \( a_n = \frac{1}{(\ln n)^{n}} \).

Apply the Root Test

The Root Test involves finding \( \limsup_{n \to \infty} \sqrt[n]{|a_n|} \). First, calculate the n-th root of the absolute value of the general term: \( \sqrt[n]{|a_n|} = \sqrt[n]{\frac{1}{(\ln n)^{n}}} = \frac{1}{\ln n} \).

Evaluate the Limit Superior

Evaluate the limit superior: \( \limsup_{n \to \infty} \frac{1}{\ln n} \). This simplifies to \( 0 \) because \( \ln n \to \infty \) as \( n \to \infty \). Thus, \( \limsup_{n \to \infty} \sqrt[n]{|a_n|} = 0 \).

Conclude with Root Test

According to the Root Test, if \( \limsup_{n \to \infty} \sqrt[n]{|a_n|} < 1 \), then the series converges. Since \( 0 < 1 \), the given series \( \sum_{n=1}^{\infty} \frac{1}{(\ln n)^{n}} \) converges.

Key concepts

Convergence

Convergence is a fundamental concept when dealing with infinite series. Essentially, when we say a series converges, we mean the sum of its terms approaches a specific finite value as more terms are added. In contrast, a series that does not settle towards a particular sum is said to diverge. There are various tests to determine whether a series converges. The Root Test is one of these methods, which is particularly useful when terms of a series involve powers of terms with logarithmic or polynomial expressions. The idea behind convergence is to ensure that adding more terms gets us closer to a certain number, rather than veering off to infinity or undefined behavior.

Series

A series is essentially the sum of the terms of a sequence. For example, in the series given in the problem, we examine an infinite sum: \[\sum_{n=1}^{\infty} \frac{1}{(\ln n)^{n}}.\]Each term in this series is given by the function, \(a_n = \frac{1}{(\ln n)^n}\). The key question is whether the sum of these terms converges or diverges. In other words, do the terms add up to a specific, finite number, or do they continue growing indefinitely? The Root Test helps in evaluating this question by investigating the behavior of the series' terms as \(n\) becomes very large.

Limit Superior

The limit superior, often denoted as \(\limsup\), is a useful tool for analyzing sequences and series. It represents the largest accumulation point of a sequence or the least upper bound over subsequential limits. When applying the Root Test, we specifically look at the \(\limsup\) of the sequence formed by the n-th roots of the absolute values of the series' terms. In this exercise, to determine the limit superior, we calculate:\[\limsup_{n \to \infty} \frac{1}{\ln n} = 0.\]As the natural logarithm \(\ln n\) tends to infinity with increasing \(n\), the reciprocal tends to zero. If the \(\limsup\) is less than 1, according to the Root Test, the series converges.

Logarithmic Functions

Logarithmic functions, such as \(\ln n\), are important in analyzing series, especially those involving exponential growth or decay. The natural logarithm, \(\ln\), grows very slowly as \(n\) increases, which can greatly affect the behavior of a series. In the given problem's series, \(\ln n\) appears in the denominator, and its growth to infinity as \(n\) increases is a critical fact in determining convergence. Understanding how logarithms behave allows us to accurately apply convergence tests like the Root Test. Essentially, for large \(n\), the influences of terms inside a logarithm diminish their impact on the convergence of a series, as seen in the outcome where the series converges as per the evaluated \(\limsup\).