Question

Use the root test to determine whether \(\sum_{m=1}^{\infty} a_{n}\) converges, where \(a_{n}\) is as follows. $$ a_{k}=\left(\frac{k-1}{2 k+3}\right)^{k} $$

Short answer

The series converges by the root test.

Step-by-step solution

Identify the Series Terms

The given series is \( \sum_{k=1}^{\infty} a_k \) where \( a_k = \left( \frac{k-1}{2k+3} \right)^k \). We need to determine if this series converges.

Apply the Root Test

For the root test, calculate \( \lim_{k \to \infty} \sqrt[k]{|a_k|} \). Substitute \( a_k = \left( \frac{k-1}{2k+3} \right)^k \) to get:\[\lim_{k \to \infty} \left( \left( \frac{k-1}{2k+3} \right)^k \right)^{1/k} = \lim_{k \to \infty} \frac{k-1}{2k+3}\]

Evaluate the Limit

Simplify \( \lim_{k \to \infty} \frac{k-1}{2k+3} \). Divide the numerator and denominator by \( k \) to get:\[\lim_{k \to \infty} \frac{1 - \frac{1}{k}}{2 + \frac{3}{k}} = \frac{1}{2}\]As \( k \to \infty \), the \( \frac{1}{k} \) terms approach zero.

Conclusion using Root Test Criterion

According to the root test, if \( \lim_{k \to \infty} \sqrt[k]{|a_k|} = L \) and \( L < 1 \), the series converges. Here, the limit is \( \frac{1}{2} \), which is less than 1, indicating that the series \( \sum_{k=1}^{\infty} a_k \) converges.

Key concepts

Understanding Series Convergence

When we talk about series convergence, we're discussing whether adding up an infinity of numbers (in this case, the terms of our series \( a_k \)) leads to a finite sum. This is an important concept in calculus because it helps us determine if a series has a well-defined sum.
In practice, to find out if a series converges, we can use a variety of tests, like the root test. This is a powerful method when the terms of the series involve expressions raised to the power of a variable.
Understanding whether a series converges or not can guide us in applications ranging from solving differential equations to analyzing electromagnetic fields. The root test matched with limit evaluation gives us a robust way to assess the behavior of series.

The Concept of Infinite Series

An infinite series is the sum of the terms of an infinite sequence. Imagine starting with the first term, adding the next, and continuing forever. This might sound bewildering, but mathematical analysis allows us to understand if this process leads to a finite number or not.
Infinite series are prevalent in various domains of mathematics and science, used to represent periodic functions (like in Fourier series) or to estimate the values of functions.
For example, consider the series given in our exercise: \( \sum_{k=1}^{\infty} a_k \). It's defined by the sequence \( a_k = \left( \frac{k-1}{2k+3} \right)^k \). This particular series makes use of terms that change with each subsequent value of \( k \), illustrating the complexity and beauty of infinite series.

Limit Evaluation in Series Convergence

Limit evaluation is at the heart of determining convergence using tests like the root test. By evaluating the limit of a function as the variable approaches infinity, we can gain insight into the behavior of the sequence or series.
In the context of the root test, calculating \( \lim_{k \to \infty} \sqrt[k]{|a_k|} \) is crucial. For our series, we derived that \( \lim_{k \to \infty} \frac{k-1}{2k+3} = \frac{1}{2}\). This limit being less than 1 is what tells us that the series converges.
When dealing with limits:

• Look for terms that simplify as \( k \to \infty \); often, these involve denominators that grow faster than numerators.
• Apply algebraic techniques like dividing terms by the highest power in the denominator, as seen in our solution steps.

Limit evaluation bridges the gap between the analysis of each term and the overall behavior of the series.