Problem 37 - Thomas Calculus

Question

Which of the series in Exercises \(15 - 48\) converge absolutely, which converge, and which diverge? Give reasons for your answers. $$ \sum _ { n = 1 } ^ { \infty } \frac { ( - 1 ) ^ { n } ( n + 1 ) ^ { n } } { ( 2 n ) ^ { n } } $$

Step-by-Step Solution

Answer

The series converges absolutely by the root test.

Step 1: Analyze the Alternating Nature

The given series is \( \sum _ { n = 1 } ^ { \infty } \frac { ( - 1 ) ^ { n } ( n + 1 ) ^ { n } } { (2 n ) ^ { n } } \). Notice that it is an alternating series due to \( (-1)^n \). This alternation affects convergence behavior.

Step 2: Apply the Absolute Convergence Test

To determine absolute convergence, consider the absolute value of the series: \( \sum _ { n = 1 } ^ { \infty } \frac { ( n + 1 ) ^ { n } } { (2 n ) ^ { n } } \). Simplify the general term: \( \frac { ( n + 1 ) ^ { n } } { (2 n ) ^ { n } } = \left( \frac{n+1}{2n} \right)^n \). This simplifies to \( \left( \frac{1}{2} + \frac{1}{2n} \right)^n \). As \( n \to \infty \), \( \left( \frac{1}{2} + \frac{1}{2n} \right)^n \to \left( \frac{1}{2} \right)^n \), which converges to zero.

Step 3: Consider the Root Test

Apply the root test to \( \left( \frac{n+1}{2n} \right)^n \). The \( n \)-th root is \( \frac{n+1}{2n} \). As \( n \to \infty \), this approaches \( \frac{1}{2} < 1 \). Since the \( n \)-th root limit is less than 1, the series converges absolutely by the root test.

Step 4: Determine the Convergence of the Non-absolute Series

Since we have confirmed absolute convergence, the original alternating series \( \sum _ { n = 1 } ^ { \infty } \frac { ( - 1 ) ^ { n } ( n + 1 ) ^ { n } } { (2 n ) ^ { n } } \) also converges. This is because absolute convergence implies convergence of the series.

Key Concepts

Alternating Series Test

The Alternating Series Test is a useful tool in determining the convergence of a series that alternates in sign. The given series \( \sum _ { n = 1 } ^ { \infty } \frac { ( - 1 ) ^ { n } ( n + 1 ) ^ { n } } { ( 2 n ) ^ { n } } \) clearly alternates due to the presence of \((-1)^n\), flipping the sign of each subsequent term. For convergence using the Alternating Series Test:

• The absolute value of the terms must be decreasing; and
• The limit of the absolute value of the terms must approach zero as \(n\) approaches infinity.

Both criteria ensure that the series converges. However, the test does not provide information about absolute convergence, a stronger form of convergence.

Absolute Convergence

Absolute convergence is a vital concept when analyzing series. A series is said to converge absolutely if the series of its absolute values converges. Consider the absolute version of our given series: \(\sum _ { n = 1 } ^ { \infty } \frac { ( n + 1 ) ^ { n } } { (2 n ) ^ { n } }\). Using the formula \(\frac { ( n + 1 ) ^ { n } } { (2 n ) ^ { n } } = \left( \frac{n+1}{2n} \right)^n\), we simplify this to \(\left( \frac{1}{2} + \frac{1}{2n} \right)^n\). As \(n \,\to \,\infty\), this tends toward \(\left( \frac{1}{2} \right)^n\).
When terms become extremely small, the series converges absolutely. Absolute convergence is crucial because if a series converges absolutely, it guarantees convergence of the original series too. This means the behavior of alternate terms does not affect the sum's convergence.

Root Test

The Root Test is another method used to determine the convergence of series, especially useful when terms involve powers of \(n\). For the series \(\sum _ { n = 1 } ^ { \infty } \left( \frac{n+1}{2n} \right)^n\), we apply the \(n\)-th root test by evaluating \(\left( \frac{n+1}{2n} \right)\).
As \(n \,\to \,\infty\), \(\frac{n+1}{2n}\) approaches \(\frac{1}{2}\), which is smaller than 1. According to the Root Test, if the limit is less than 1, the series converges absolutely. This provides a clear pathway to establish convergence even if other simpler tests, like the Direct Comparison Test, are cumbersome to apply.

Series Divergence

Sometimes it might be necessary to identify when a series diverges, which is the opposite of convergence. Divergence in series often happens when the partial sums of the series do not approach any finite limit as \(n\) becomes large. In simpler words, the series sum keeps increasing without ever settling down.

• A series fails to meet the convergence criteria when testing with tools like the Alternating Series Test or the Root Test.
• A limit of the general term does not approach zero, signaling possible divergence.

Recognizing divergence early helps avoid repetitive calculations in checking every possible test for convergence.

Mathematical Analysis

Mathematical Analysis provides the rigorous foundation for understanding the convergence behavior of series. It goes beyond checking a series with tests, delving into why these tests work based on behavior patterns and limits.

• It involves understanding limits, continuity, and differentiability.
• Mathematical Analysis allows for exploring the implications of convergence or divergence in deeper mathematical contexts, such as function approximation and complex integrations.

Through such analysis, mathematicians build confidence in results by linking individual steps to a broader spectrum of findings. It gives clarity to why results hold true, supporting a student's journey in mastering higher-level math concepts.