Question

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

Short answer

The series \( \sum_{k=1}^{\infty} a_k \) diverges by the root test.

Step-by-step solution

Apply the Root Test Formula

The root test involves taking the nth root of the absolute value of the sequence term and then examining the limit. For a series \( \sum a_k \), apply the root test by calculating \( \lim_{k \to \infty} \sqrt[k]{|a_k|} \). In this case, we have \( a_k = \frac{k^k}{e^k} \), so \( \sqrt[k]{|a_k|} = \left( \frac{k^k}{e^k} \right)^{1/k} = \frac{k}{e} \).

Simplify the Root

Simplifying further, we get \( \lim_{k \to \infty} \sqrt[k]{|a_k|} = \lim_{k \to \infty} \frac{k}{e} \). The term \( \frac{k}{e} \) tends to infinity as \( k \to \infty \).

Determine Convergence Using the Root Test

According to the root test, if \( \lim_{k \to \infty} \sqrt[k]{|a_k|} > 1 \), the series diverges. Since \( \lim_{k \to \infty} \frac{k}{e} = \infty \) which is greater than 1, it indicates that the series \( \sum_{k=1}^{\infty} a_k \) diverges.

Key concepts

Convergence

Convergence of a series is a central concept in calculus and analysis. It describes whether the sum of a series approaches a finite number as we add more and more terms. In mathematical terms, a series \( \sum_{n=1}^{\infty} a_n \) converges if its partial sums \( S_n = a_1 + a_2 + \dots + a_n \) approach a specific limit \( L \) as \( n \to \infty \). This limit \( L \) is the value that the series is said to converge to.

For a series to converge, the terms of the series must decrease to zero; however, this alone isn't sufficient for convergence. The choice of test to determine convergence depends on the form of \( a_n \). The root test, for instance, is particularly useful when \( a_n \) includes exponential terms, like \( \frac{k^k}{e^k} \), since it involves taking roots which can simplify such expressions.

Understanding convergence helps in differentiating series that have well-defined sums from those that do not, informing us about their behavior over infinite processes.

Series Divergence

Series divergence asserts that the sum of the terms of a series does not approach a finite number as more terms are added, meaning it trends towards infinity, oscillates, or doesn't settle around any particular value at all. A divergent series can arise in various forms, and knowing that a series diverges can be as important as knowing it converges.

When applying the root test to the given series \( \sum_{k=1}^{\infty} a_k \), we find that the root of the expression tends to infinity. Specifically, \( \lim_{k \to \infty} \sqrt[k]{|a_k|} = \lim_{k \to \infty} \frac{k}{e} = \infty \). Because this limit is greater than 1, it indicates divergence according to root test criteria.

It is crucial to remember that divergence implies that partial sums of the series grow without bound. Recognizing divergence through tests like the root test prevents futile efforts in trying to assign a value to something that inherently does not possess one.

Limit Evaluation

Limit evaluation is a vital technique for analyzing the behavior of functions and sequences as they approach a certain point, often infinity. In the context of the root test, the limit we evaluate determines convergence or divergence of a series. Specifically, we look at \( \lim_{k \to \infty} \sqrt[k]{|a_k|} \).

For the sequence \( a_k = \frac{k^k}{e^k} \), the root simplifies to \( \frac{k}{e} \). Evaluating this limit as \( k \to \infty \), we notice it steadily increases without bound. This information directly influences our conclusion about the series' behavior.

Evaluating limits effectively requires understanding the interaction of functions involved, such as exponential growth or decay. Here, recognizing that \( k^k \) grows much faster than \( e^k \), explains why \( \frac{k}{e} \) tends towards infinity. Every step in limit evaluation enhances our capacity to predict the ultimate outcome of the series.