Problem 8 - Basic Analysis: Introduction to Real Analysis

Question

Show that \(\sum_{n=1}^{\infty} \frac{(-1)^{n}}{n}\) converges. Hint: Consider the sum of two subsequent entries.

Step-by-Step Solution

Answer

The series converges by the Alternating Series Test.

Step 1: Identify the Series

The series given is \( S = \sum_{n=1}^{\infty} \frac{(-1)^{n}}{n} \). This is an alternating series due to the presence of \((-1)^n\).

Step 2: Apply the Alternating Series Test

The Alternating Series Test states that the series \( \sum (-1)^n a_n \) converges if \( a_n \) is positive, decreasing, and approaches zero as \( n \to \infty \). Here, \( a_n = \frac{1}{n} \).

Step 3: Check if \( a_n > 0 \)

Clearly, \( \frac{1}{n} > 0 \) for all \( n \geq 1 \). Thus, the condition \( a_n > 0 \) is satisfied.

Step 4: Verify \( a_n \to 0 \) as \( n \to \infty \)

Since \( a_n = \frac{1}{n} \), we see \( \lim_{n \to \infty} \frac{1}{n} = 0 \). This condition is also fulfilled.

Step 5: Show \( a_n \) is Decreasing

To show \( \{a_n\} \) is decreasing, check if \( a_{n+1} < a_n \). Given \( \frac{1}{n+1} < \frac{1}{n} \), the sequence is decreasing for all \( n \geq 1 \).

Step 6: Conclude Convergence

Since all conditions of the Alternating Series Test are satisfied, the series \( \sum_{n=1}^{\infty} \frac{(-1)^n}{n} \) converges.

Key Concepts

Convergence of Series

In mathematics, the concept of the convergence of a series is fundamental. When we talk about a series converging, it simply means that as we keep adding more terms, the total sum approaches a specific value. There are many types of convergence tests, with each test offering a different perspective on determining if a series will converge. For instance, geometric series and p-series come with specific rules that allow us to judge their convergence easily.

In the case of an alternating series such as \( \sum_{n=1}^{\infty} \frac{(-1)^n}{n} \), we use the Alternating Series Test to determine convergence. Understanding the conditions under which a series converges is crucial, especially when dealing with infinite series in mathematical analysis. Real-life applications include solving differential equations, signal processing, and in physics, where series can approximate various functions or quantities. Thus, the convergence tells us that the series can be useful for calculations or accurate approximations in practical scenarios.

Real Analysis

Real analysis is a branch of mathematics that deals with real numbers and the functions of real variables. It involves rigorous treatments of the convergence of sequences and series, among other fundamental concepts. Real analysis builds the foundation for differentiating and integrating functions, which are critical for scientific investigations.

Part of studying real analysis includes understanding why certain series converge and others do not. The Alternating Series Test explained in the step-by-step solution is a tool derived from real analysis theories that allow mathematicians to analyze functions and series involving alternating elements like \((-1)^n\). This approach is systematic and helps break down more complex mathematical phenomena into understandable concepts.

Applications of real analysis extend beyond pure mathematics, enabling advancements in platform-specific calculations in engineering, computer science, and economics, where precise calculations are essential.

Alternating Series

An alternating series is one where the signs of the terms alternate between positive and negative as we progress from one term to the next. A perfect example is the series \( \sum_{n=1}^{\infty} \frac{(-1)^n}{n} \), as given in the exercise. This type of series is notable for its unique properties and behaviors, which set it apart from non-alternating series.

To test the convergence of an alternating series, we use the Alternating Series Test, which requires that the terms \( a_n \) are positive, decreasing, and approach zero as \( n \) tends to infinity. For instance, if \( a_n = \frac{1}{n} \), as shown, the series converges because it meets all the necessary conditions. However, if any of these conditions fail, convergence cannot be assured.

This test provides a reliable method to determine the convergence of alternating series, which are common in many areas, including Fourier analysis and various summation problems in physics and engineering. Appreciating how alternating series function can greatly enhance one's understanding of how infinite sequences sum to finite values.