Question

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

Short answer

The series \(\sum_{m=1}^{\infty} \frac{n}{e^n}\) converges by the root test.

Step-by-step solution

Write the Root Test Formula

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 \(L < 1\) and diverges if \(L > 1\). It is inconclusive if \(L = 1\).

Substitute the Given Term

The term \(a_n = \frac{n}{e^n}\). We need to find \(\lim_{n \to \infty} \sqrt[n]{\left|\frac{n}{e^n}\right|}\) which simplifies to \(\lim_{n \to \infty} \frac{n^{1/n}}{e}\).

Analyze \(n^{1/n}\)

Recall that \(\lim_{n \to \infty} n^{1/n} = 1\). As \(n\) grows large, \(n^{1/n}\) approaches 1. Thus, we have \(\lim_{n \to \infty} \frac{n^{1/n}}{e} = \frac{1}{e}\).

Evaluate the Limit and Apply the Root Test

Since \(\lim_{n \to \infty} \frac{1}{e} = \frac{1}{e}\), which is less than 1, the root test confirms that the series \(\sum_{n=1}^{\infty} a_n\) converges.

Key concepts

Convergence of Infinite Series

Infinite series are sums that continue indefinitely. Understanding when these series converge is important in mathematical analysis. Convergence means that as you keep adding more and more terms, the sum approaches a fixed number. When dealing with infinite series, it's crucial to determine whether or not this convergence occurs. The series converges if the sum of its terms gets infinitely close to a specific number as more terms are added.

For example, with the series given by \(a_n = \frac{n}{e^n}\), we use tests like the root test to determine convergence. If an infinite series converges, it implies stability and predictability in mathematical models or real-world applications such as financial predictions or physical phenomena simulations.

In this context, convergence provides assurance that the calculations will have meaningful and reliable results. It's also worth noting the distinction between convergence and divergence. If a series does not converge, its sum might increase indefinitely or oscillate without settling at a particular value.

Limit Evaluation

Evaluating limits is a powerful tool in calculus, especially for analyzing sequences and series. The concept of a limit helps us understand the behavior of functions as they get close to a particular point or as their input grows large. For infinite series, such as \(\frac{n}{e^n}\), evaluating limits can reveal the growth tendencies of the series' terms.

In our example, we use the limit \(\lim_{n \to \infty} \frac{n^{1/n}}{e}\) as part of the root test to determine convergence. Calculating the limit requires understanding how numerator and denominator behave as \(n\) increases.

For \(n^{1/n}\), we remember that as \(n\) goes to infinity, this expression approaches 1. Limit evaluations like this simplify complex expression behaviors, thus making it easier to apply tests and determine convergence or divergence. Swapping complex expressions for their limits makes analytical work more manageable and insightful.

Exponential Functions

Exponential functions, especially of the form \(e^x\), play a vital role in mathematical series and calculus in general. An exponential function grows very rapidly compared to other polynomial functions, which have a significant impact on series behavior.

In our problem, \(a_n = \frac{n}{e^n}\) involves an exponential function \(e^n\). The exponential term in the denominator grows much faster than the linear term \(n\) in the numerator. This rapid growth is crucial in the root test for convergence.

Understanding the behavior of exponential functions helps with series analysis, particularly in determining whether the series converges or diverges. In mathematical applications, exponential functions model diverse phenomena ranging from population growth to radioactive decay, making a deep understanding of their properties essential.