Problem 15 - Thomas Calculus

Question

In Exercises \(9-16,\) use the Root Test to determine if each series converges absolutely or diverges. $$\sum_{n=1}^{\infty}(-1)^{n}\left(1-\frac{1}{n}\right)^{n^{2}}$$ $$(Hint: \lim _{n \rightarrow \infty}(1+x / n)^{n}=e^{x} )$$

Step-by-Step Solution

Answer

The series converges absolutely by the Root Test.

Step 1: Identify the terms of the series

The terms of the series are given by \((-1)^{n} \left(1 - \frac{1}{n}\right)^{n^{2}}\). We are interested in using the Root Test to determine the convergence.

Step 2: Apply the Root Test

To apply the Root Test, we compute \( \lim_{n \to \infty} \sqrt[n]{|a_n|} \) where \( a_n = (-1)^{n} \left(1 - \frac{1}{n} \right)^{n^{2}} \). Since \((-1)^n\) does not affect the absolute value, we have \(|a_n| = \left(1 - \frac{1}{n}\right)^{n^2}\).

Step 3: Simplify the expression inside the limit

We need to evaluate \( \lim_{n \to \infty} \sqrt[n]{\left(1 - \frac{1}{n}\right)^{n^2}} = \lim_{n \to \infty} \left(1 - \frac{1}{n}\right)^{n^2/n} = \lim_{n \to \infty} \left(1 - \frac{1}{n}\right)^{n}\).

Step 4: Use the hint to evaluate the limit

Using the hint \( \lim_{n \to \infty} \left(1 + \frac{x}{n}\right)^{n} = e^{x} \), we substitute \( x = -1 \) to find \(\lim_{n \to \infty} \left(1 - \frac{1}{n}\right)^{n} = e^{-1} = \frac{1}{e} \).

Step 5: Interpret the result of the Root Test

The Root Test states that if \( \lim_{n \to \infty} \sqrt[n]{|a_n|} = L < 1 \), the series converges absolutely. In our case, \( \frac{1}{e} < 1 \), therefore, the series converges absolutely.

Key Concepts

Series Convergence

Series convergence is a concept that helps determine if a series approaches a finite value as more terms are added. In mathematics, a series is the sum of the terms of a sequence. Convergence means that as we add more and more terms, the series approaches a specific, finite value.

To evaluate whether a series converges, various tests can be applied. In the problem mentioned, we use the Root Test. This test is particularly useful for a series with terms involving exponents and for identifying absolute convergence.

The Root Test considers the limit of the nth root of the absolute value of the terms in the series. If this limit results in a value less than 1, it indicates that the series converges absolutely. If it equals 1, the test is inconclusive. If it is greater than 1, the series diverges.

Absolute Convergence

Absolute convergence is a stronger form of convergence for series. If a series converges absolutely, it means the series will converge even when all its terms are replaced with their absolute values. This form of convergence implies that the series is very stable.

Absolute convergence is especially significant because it allows us to manipulate the series more freely, such as changing the order of the terms without affecting the sum. This is not possible for conditionally convergent series. By applying the Root Test in our exercise, we determined that the series converges absolutely since the computed limit was less than 1.

• If \(|a_n|\ ightarrow 0\) and the absolute series converges, the original series converges absolutely.
• The ability to reorder terms or conduct analysis without altering the result is a valuable property of absolutely convergent series.

Limit Evaluation

Evaluating limits is a critical part of applying the Root Test. In our exercise, we needed to find the limit as n approaches infinity of the nth root of the absolute value of the terms.

The given exercise provided a hint about the limit of the sequence \(\left( 1 + \frac{x}{n} \right)^n\), which helps to simplify our calculations. This concept is rooted in Euler’s number, \(e\), a pivotal constant in mathematics. The hint helps us approximate such expressions using \(e^{-1}\) for the given problem.

• Understanding and easily computing limits can simplify seemingly complex expressions.
• Limits are foundational in calculus, especially when dealing with infinities and continuous functions.