Question

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

Short answer

The series converges.

Step-by-step solution

Define the Series Terms

The series we need to analyze is given by \(a_n = \left(\frac{1}{e} + \frac{1}{n}\right)^n\). Our task is to determine the convergence of \(\sum_{n=1}^{\infty} a_n\) using the root test.

Apply the Root Test

The root test states that for a series \(\sum_{n=1}^{\infty} a_n\), we compute the following limit: \(L = \lim_{n \to \infty} \sqrt[n]{|a_n|}\). If \(L < 1\), the series converges; if \(L > 1\), the series diverges; if \(L = 1\), the test is inconclusive.

Compute the Root

For \(a_n = \left(\frac{1}{e} + \frac{1}{n}\right)^n\), we first take the nth root: \(\sqrt[n]{a_n} = \left(\frac{1}{e} + \frac{1}{n}\right)\). So, the limit we need to compute is \(L = \lim_{n \to \infty} \left(\frac{1}{e} + \frac{1}{n}\right)\).

Evaluate the Limit

Calculate the limit \(L = \lim_{n \to \infty} \left(\frac{1}{e} + \frac{1}{n}\right) = \frac{1}{e}\), because the term \(\frac{1}{n}\) approaches zero as \(n\) approaches infinity. Thus, \(L = \frac{1}{e}\).

Conclusion of the Root Test

Since \(L = \frac{1}{e} < 1\), according to the root test, the series \(\sum_{n=1}^{\infty} a_n\) converges.

Key concepts

Root Test

The root test is a powerful method to determine the convergence of an infinite series. It involves analyzing the nth root of the absolute value of the series' terms. Specifically, you calculate the limit \( L = \lim_{n \to \infty} \sqrt[n]{|a_n|} \). Based on this, we use the following criteria:

• If \( L < 1 \), the series converges.
• If \( L > 1 \), the series diverges.
• If \( L = 1 \), the test is inconclusive.

To apply the root test, it's crucial to find an expression for \( \sqrt[n]{|a_n|} \) and then compute its limit as \( n \) approaches infinity. This helps determine if the series grows or diminishes effectively enough to converge.

Infinite Series

An infinite series is the sum of an infinite sequence of numbers. It's represented as \( \sum_{n=1}^{\infty} a_n \), where \( a_n \) denotes the nth term of the series. The concept of convergence is central here, as it tells us whether we can assign a finite value to the endless summation.In the original exercise, the series defined is \( \sum_{n=1}^{\infty} a_n \), with \( a_n = \left(\frac{1}{e} + \frac{1}{n}\right)^n \). Each term combines fractions and exponents, presenting more complexity in analyzing its long-term behavior.Understanding infinite series involves breaking down the sequence's terms and using various tests, like the root test, to gauge whether they accumulate to a finite total or not.

Convergence Criteria

Convergence criteria are rules or tests that help determine whether an infinite series converges to a finite value or tends toward infinity. These criteria are vital in mathematical analysis as they offer systematic approaches to assess the behavior of series in the infinite realm.When applying convergence criteria, such as the root test, we focus on specific characteristics of the series terms. For example, analyzing \( \sqrt[n]{a_n} = \left(\frac{1}{e} + \frac{1}{n}\right) \) provided a clear path to apply the limit test. As \( n \) grows indefinitely, the term \( \frac{1}{n} \) trends to zero, simplifying our limit calculation to \( L = \frac{1}{e} \).The result of \( L < 1 \) confirmed convergence in this scenario, showing the effectiveness of relying on mathematical criteria to draw conclusions about infinite sums.