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. $$\sum_{n=1}^{\infty} \frac{n^{2}}{(\ln n)^{n}}$$

Short answer

The series converges by the Root Test since the limit is 0.

Step-by-step solution

Identify the General Term

For the series \( \sum_{n=1}^{\infty} \frac{n^{2}}{(\ln n)^{n}} \), the general term \( a_n \) is \( a_n = \frac{n^{2}}{(\ln n)^{n}} \). We'll use this to apply the Root Test.

Apply the Root Test

The Root Test involves calculating \( \lim_{n \to \infty} \sqrt[n]{|a_n|} \). For our series, this is \[ \lim_{n \to \infty} \sqrt[n]{\frac{n^{2}}{(\ln n)^{n}}} = \lim_{n \to \infty} \frac{\sqrt[n]{n^{2}}}{\ln n}. \]

Simplify the Expression

Notice that \( \sqrt[n]{n^{2}} = (n^2)^{1/n} = n^{2/n} = e^{2 \ln n / n} \). As \( n \to \infty \), \( e^{2 \ln n / n} \to e^{0} = 1 \). Thus, \( \sqrt[n]{n^{2}} \to 1 \) as \( n \to \infty \).

Evaluate the Limit

Substituting back, we have \[ \lim_{n \to \infty} \frac{1}{\ln n} = 0. \] Thus, \( \lim_{n \to \infty} \sqrt[n]{|a_n|} = 0 \).

Assess the Convergence

Since \( \lim_{n \to \infty} \sqrt[n]{|a_n|} = 0 < 1 \), by the Root Test, the series \( \sum_{n=1}^{\infty} \frac{n^{2}}{(\ln n)^{n}} \) converges.

Key concepts

Series Convergence

Series convergence is a fundamental concept in calculus that determines whether an infinite series will add up to a finite number. To determine convergence, mathematicians apply various tests based on the series' nature. In our example, we use the Root Test to analyze the sum \( \sum_{n=1}^{\infty} \frac{n^{2}}{(\ln n)^{n}} \). Here, convergence means if you add up the terms of the series, the total gets closer and closer to a definite sum.Convergence tests like the Root Test help determine if an infinite series is convergent or divergent. If a series is convergent, as we add more and more terms, the sum approaches a specific value. For this series, since applying the Root Test yielded a limit less than one, it was confirmed to be convergent.

Limit Evaluation

Evaluating limits is a powerful tool used to determine the behavior of sequences and series, especially as a variable approaches infinity. In applying the Root Test, evaluating the limit \( \lim_{n \to \infty} \sqrt[n]{|a_n|} \) tells us how the terms of the series behave as \( n \) becomes very large.The given series expresses a general term where the behavior of \( \frac{1}{\ln n} \) as \( n \to \infty \) was crucial for limit evaluation:

• The expression \( \sqrt[n]{n^2} \) was evaluated to approach 1, as \( n \to \infty \).
• Then, \( \lim_{n \to \infty} \frac{1}{\ln n} = 0 \) showcases that the limit of the sequence is slowly approaching zero.

Understanding the limit evaluation is essential, as it provides insight into why the Root Test concluded the series was convergent: because the evaluated limit was less than one.

General Term

The general term of a series is a formula that describes how to find any term in the sequence from its position number. For the series \( \sum_{n=1}^{\infty} \frac{n^{2}}{(\ln n)^{n}} \), the general term \( a_n \) is given by \( a_n = \frac{n^{2}}{(\ln n)^{n}} \).The general term is a cornerstone aspect because it is used to apply convergence tests like the Root Test. Knowing \( a_n \) allows you to:

• Describe every term of the series.
• Evaluate limits necessary for determining the series' convergence.
• Understand how each term's size and behavior change as \( n \) increases.

In our problem, identifying and manipulating the general term was crucial for applying the Root Test. It helped simplify expressions and allowed precise limit evaluations to conclude on the series' convergence.