Question

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

Short answer

The series diverges by the root test.

Step-by-step solution

Understand the Root Test Setup

The root test is used to determine the convergence of an infinite series \( \sum_{n=1}^{\infty} a_n \). According to the root test, if \( \lim_{n \to \infty} \sqrt[n]{|a_n|} = L \) where \( L < 1 \), then the series converges absolutely. If \( L > 1 \) or \( L = \infty \), then the series diverges. If \( L = 1 \), the test is inconclusive.

Identify the Terms

Identify the terms given in the series, which in this case is \( a_k = \frac{\pi^k}{k^2} \). This will be used in the root test formula.

Apply the Root Test Formula

Calculate \( \lim_{k \to \infty} \sqrt[k]{|\frac{\pi^k}{k^2}|} \). This requires the simplification of the term under the k-th root.

Simplify and Compute the Limit

Simplify \( \sqrt[k]{|\frac{\pi^k}{k^2}|} \) as follows:\[\sqrt[k]{|\frac{\pi^k}{k^2}|} = \left(\frac{\pi^k}{k^2}\right)^{\frac{1}{k}} = \frac{\pi}{k^{2/k}}\]Now compute \( \lim_{k \to \infty} \frac{\pi}{k^{2/k}} \). Recognize that \( k^{2/k} \to 1 \) as \( k \to \infty \). Thus, \( \frac{\pi}{k^{2/k}} \to \pi \).

Interpret the Result

Since \( \lim_{k \to \infty} \sqrt[k]{|a_k|} = \pi \), and \( \pi > 1 \), according to the root test, the series \( \sum_{k=1}^{\infty} a_k \) diverges. The root test shows \( L > 1 \).

Key concepts

Convergence of Infinite Series

In mathematics, we often deal with infinite series, which are sums of infinite sequences of numbers. The convergence of such series is crucial in determining if a meaningful sum exists. A series converges when its terms approach a specific value, providing a finite sum. This concept is fundamental in analysis, helping mathematicians understand series behavior over an infinite scope. To determine the convergence of an infinite series, various tests can be applied, like the Root Test. When a series converges, it means that no matter how far you progress through its terms, their cumulative sum gets closer to a specific value, never wildly fluctuating or heading towards infinity. This concept helps ensure that processes modeled by infinite series have stable results. In applying convergence tests, identifying the type of series and conditions for each test is essential. For instance, the Root Test is specifically suited for series where terms become less influential in magnitude quickly enough.

Limit Computation

Limit computation involves finding the value a function approaches as the input approaches a certain point. In the context of the Root Test, we compute the limit of the nth root of the absolute value of the term of the series as it approaches infinity. Understanding this requires familiarity with limits and their properties. It helps evaluate how series terms behave relative to each other as they approach infinity. For the series given, we calculate the limit:\[\lim_{k \to \infty} \sqrt[k]{|a_k|} = \lim_{k \to \infty} \frac{\pi}{k^{2/k}}\]Here, recognizing properties of limits, like \[k^{2/k} \to 1\] as \(k \to \infty\) becomes crucial. This sophisticated calculation method informs the series' convergence or divergence. By identifying the trend of individual terms and their effect on the series sum, limits play a pivotal role in interpreting results from tests like the Root Test.

Divergence

Divergence, the opposite of convergence, happens when a series does not settle on a limit. This could mean the series terms grow larger indefinitely or fluctuate without approaching a specific value. Recognizing divergence is essential to determine when a series can be deemed not to have a finite sum. In our context, the limit we found for \(\lim_{k \to \infty} \sqrt[k]{|a_k|} = \pi\) shows that the limit exceeds one. According to the Root Test criterion, when \(L > 1\) or \(L = \infty\), the series diverges. Here, the terms don't stabilize around a finite sum, indicating that the series' partial sums could become infinitely large. Knowing when a series diverges prevents incorrect assumptions about potential infinite sums and ensures proper analysis is applied.

Infinite Series

An infinite series is essentially the sum of an infinite sequence of terms. While individual terms in a series can be small, their infinite sum may lead to a large, finite, or undefined result, depending on the series nature. Evaluating infinite series requires checking whether their total adds up or stretches towards infinity. The series in focus here involves terms like \(a_k = \frac{\pi^k}{k^2}\), which are structured in a form that benefits from applying the Root Test. This test, ideal for such exponentially growing terms, quickly reveals whether they converge or diverge. In practical terms, understanding infinite series helps in evaluating complex processes in fields like physics and engineering, where truly infinite sums are idealized concepts necessary for modeling real-world phenomena. Through mathematical frameworks, we decide if an infinite series gives meaningful information or if its infinite stretch impacts the outcomes negatively, like in the case of divergence.