Question

Use the root test to determine whether \(\sum_{m=1}^{\infty} a_{n}\) converges, where \(a_{n}\) is as follows. $$ a_{k}=\frac{1}{(1+\ln k)^{k}} $$

Short answer

The series converges by the root test.

Step-by-step solution

Understand the Root Test

The root test states that for a series \( \sum_{n=1}^{\infty} a_n \), if \( \lim_{n \to \infty} \sqrt[n]{|a_n|} = L \), then the series converges if \( L < 1 \), diverges if \( L > 1 \), and is inconclusive if \( L = 1 \). We apply this test to our series.

Compute \(a_k^{1/k}\)

Given \( a_k = \frac{1}{(1+\ln k)^{k}} \), we need \( \left( \frac{1}{(1+\ln k)^{k}} \right)^{1/k} = \frac{1}{1+\ln k} \).

Take the Limit

Find the limit \( \lim_{k \to \infty} \frac{1}{1+\ln k} \). As \( k \to \infty \), \( \ln k \to \infty \), making \( 1+\ln k \to \infty \).Thus, \( \frac{1}{1+\ln k} \to 0 \).

Apply Test Results

Since \( \lim_{k \to \infty} \sqrt[k]{a_k} = 0 \), and \( L = 0 < 1 \), the series \( \sum_{k=1}^{\infty} a_k \) converges by the root test.

Key concepts

Understanding the Root Test

The root test is a powerful tool in calculus to determine the convergence of infinite series. It is particularly useful when dealing with series where each term is raised to a power involving the variable, such as our provided series.
When applying the root test for a series \( \sum_{n=1}^{\infty} a_n \), the key step is to compute the limit \( \lim_{n \to \infty} \sqrt[n]{|a_n|} \). Here's the crucial part:

• If this limit \( L \) is less than 1 \((L < 1)\), the series converges.
• If \( L \) is greater than 1 \((L > 1)\), the series diverges.
• If \( L \) equals 1 \((L = 1)\), the test is inconclusive.

In our exercise, we are tasked with applying the root test to determine the convergence of a specific series. We transform the expression using the root test criterion by calculating the \( k^{th} \) root of the term \( a_k \). This simplification is the heart of the root test, turning a complex series into a simpler expression to evaluate. Once simplified, the final step is evaluating the limit of this expression as \( n \) (or \( k \) in this case) approaches infinity.

Series Convergence Explained

A series is essentially a way of adding an infinite sequence of numbers, and understanding whether these sums approach a finite value is crucial. This concept is what we refer to as convergence.
A series converges if the sum of its terms approaches a specific number as you add more and more terms. If this sum becomes infinitely large or doesn't settle on a fixed number, the series diverges. In our context:

• The root test helps determine series convergence by evaluating the behavior of terms raised to their reciprocal power.
• Convergence implies that adding more terms will still result in a value close to this fixed sum.
• Divergence indicates the series does not have a sum that can be pinned down to a single value.

Using these criteria, the exercise revealed that our series indeed converges because the application of the root test resulted in a limit \( L = 0 \), clearly less than 1.

Role of Logarithmic Functions

Logarithmic functions, like \( \ln k \), play a significant part in our series convergence problem. Logarithms help us understand growth rates and are commonly used in analyzing series due to their fundamental properties. Let's explore:

• In our expression \( a_k = \frac{1}{(1+\ln k)^k} \), the \( \ln k \) affects the base of the power heavily.
• Logarithms grow slower than linear functions; thus, as \( k \to \infty \), \( \ln k \) will increase at a decreasing rate.
• This helps the denominator in our series expression \((1+\ln k)^k\) grow very large, contributing significantly to the convergence of the series.

Understanding how logarithms control the growth rate of terms is essential to see why the root test yields \( L = 0 \). This insight shows us why the series converges as \( k \) increases, with \( 1+\ln k \) growing to infinity, making the reciprocal term \( \frac{1}{1+\ln k} \) shrink toward zero.