Question

In the following exercises, use either the ratio test or the root test as appropriate to determine whether the series \(\sum a_{k}\) with given terms \(a_{k}\) converges, or state if the test is inconclusive. $$ a_{n}=\left(n^{1 / n}-1\right)^{n} $$

Short answer

The series \( \sum \left( n^{1/n} - 1 \right)^n \) converges.

Step-by-step solution

Analyze the given series

The series we need to analyze is given by the term \( a_n = \left( n^{1/n} - 1 \right)^n \). Our goal is to determine whether this series converges or not.

Choose the appropriate test

Given the form of \( a_n = \left( n^{1/n} - 1 \right)^n \), it involves powers, which suggests that the root test will be more appropriate for this series.

Apply the root test

For the root test, we define \( L = \lim_{n \to \infty} \sqrt[n]{|a_n|} \), and check the value of \( L \). If \( L < 1 \), the series converges; if \( L > 1 \), the series diverges; and if \( L = 1 \), the test is inconclusive.

Simplify the root term

Calculate \( \sqrt[n]{a_n} = \sqrt[n]{(n^{1/n} - 1)^n} = n^{1/n} - 1 \). Thus, the limit \( L = \lim_{n \to \infty} (n^{1/n} - 1) \) needs to be evaluated.

Evaluate the limit

As \( n \to \infty \), notice that \( n^{1/n} \to 1 \). Thus, \( n^{1/n} - 1 \to 0 \). Therefore, \( L = \lim_{n \to \infty} (n^{1/n} - 1) = 0 \).

Conclusion from the root test result

Since \( L = 0 < 1 \), according to the root test, the series \( \sum a_n \) converges.

Key concepts

Root Test

The root test is a powerful tool in determining the convergence of a series, especially when terms involve exponents. It's particularly useful for series where each term has a form like \(a_k^n\). To use the root test, given a series \(\sum a_n\), we consider the sequence formed by the n-th root of the absolute value of the terms: \(L = \lim_{n \to \infty} \sqrt[n]{|a_n|}\).

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

The root test is valuable when you have a series like \(a_n = \left( n^{1/n} - 1 \right)^n\) because it simplifies the process of evaluating the convergence behavior by focusing directly on the limit of the sequence.

Ratio Test

The ratio test is another essential convergence tool that's best applied to series with factorials or exponential expressions. For any series \(\sum a_n\), we find \(\lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| = L\). Just like the root test:

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

The ratio test is particularly useful for dealing with series where terms grow or shrink rapidly, such as geometric series or series involving factorial growth. However, it is not applicable for every series, such as those involving roots as seen in \(a_n = \left( n^{1/n} - 1 \right)^n\), where the root test is more appropriate.

Series Convergence

Series convergence is a fundamental concept in calculus that determines whether the sum of an infinite series approaches a finite limit. A series \(\sum a_n\) is convergent if the sequence of partial sums \(S_n = a_1 + a_2 + a_3 + \ldots + a_n\) approaches a specific number as \(n \rightarrow \infty\). If a series does not converge, it is said to diverge.

• A common method to determine convergence is the comparison test.
• The root and ratio tests are usually applied when the series involves exponential growth or decay.
• For telescoping series, partial fraction decomposition is often useful.

Convergence is crucial because it ensures that the infinite sum yields a practical, finite result, which is often necessary for solving real-world problems.

Limit Evaluation

Limit evaluation is a technique used to determine the behavior of sequences and series as they extend indefinitely. When we analyze a series \(\sum a_n\), limits help us understand if the terms are trending towards a specific value as \(n\) becomes very large. In the root test, this involves evaluating \(L = \lim_{n \to \infty} \sqrt[n]{|a_n|}\).There are standard limit properties and techniques, such as L'Hôpital's Rule, which can deal with indeterminate forms or simplify complex limits.

• Direct substitution is used when the limit function is continuous at the limit point.
• L'Hôpital's Rule applies when limits yield forms like \(\frac{0}{0}\) or \(\frac{\infty}{\infty}\).

For the series \(a_n = \left( n^{1/n} - 1 \right)^n\), limit evaluation shows \(\lim_{n \to \infty} (n^{1/n} - 1) = 0\), which confirms the series converges as per the root test.