Problem 35 - Thomas Calculus

Question

Which of the series converge, and which diverge? Use any method, and give reasons for your answers. \begin{equation} \sum_{n=1}^{\infty} \frac{1-n}{n 2^{n}} \end{equation}

Step-by-Step Solution

Answer

The series \( \sum_{n=1}^{\infty} \frac{1-n}{n2^n} \) converges by the Limit Comparison Test.

Step 1: Analyze the Series

The given series is \( \sum_{n=1}^{\infty} \frac{1-n}{n2^n} \). This is an infinite series with terms of the form \( a_n = \frac{1-n}{n2^n} \). We need to determine if this series converges or diverges.

Step 2: Apply the Limit Comparison Test

To apply the Limit Comparison Test, we compare the given series with a known series. A good choice is \( \sum \frac{1}{2^n} \), a convergent geometric series. Calculate the limit: \[ \lim_{n \to \infty} \frac{\frac{1-n}{n2^n}}{\frac{1}{2^n}} = \lim_{n \to \infty} \frac{1-n}{n} = \lim_{n \to \infty} \frac{1}{n} - 1 = -1. \]This limit is finite, which indicates that the series \( \sum_{n=1}^{\infty} \frac{1-n}{n 2^n} \) and \( \sum_{n=1}^{\infty} \frac{1}{2^n} \) have similar convergence behavior.

Step 3: Conclude Convergence Based on the Comparison

Since \( \sum \frac{1}{2^n} \) is convergent and the Limit Comparison Test gives us a non-zero finite limit, \( \sum_{n=1}^{\infty} \frac{1-n}{n 2^n} \) converges as well.

Key Concepts

Infinite Series

An infinite series is a sum of an infinite sequence of terms. Considering the given series, it is expressed as \( \sum_{n=1}^{\infty} \frac{1-n}{n 2^n} \). When working with infinite series, one of the main concerns is determining whether the sum converges to a finite number or diverges. Convergence means that as we add more and more terms of the series, the sum approaches a specific finite value, whereas divergence indicates that the sum either increases indefinitely or oscillates without approaching a fixed limit. Understanding the behavior of an infinite series often requires looking at the nature of its terms and possibly comparing it to other known series. This is where various tests for convergence come into play, providing mathematicians and students alike with tools to analyze the properties of these series.

Limit Comparison Test

The Limit Comparison Test is a tool used to determine the convergence or divergence of an infinite series by comparing it to another series with known behavior. To apply this test, you take the limit of the ratio of the terms of your series and another series. If the limit is a non-zero finite number, both series either converge or diverge together. In the solution, the series \( \sum_{n=1}^{\infty} \frac{1-n}{n2^n} \) was compared to the series \( \sum_{n=1}^{\infty} \frac{1}{2^n} \), a geometric series known to converge. Calculating the limit of the ratio between the terms of these two series gave a finite limit of \( -1 \). Despite the negative number, this non-zero finite result tells us that both series will have the same convergence behavior. Since the geometric series is convergent, so too is the given series. Using the Limit Comparison Test can be particularly handy when the series in question is complex, allowing for a breakdown of its behavior compared to simpler, known series.

Geometric Series

A geometric series is a series where each term is a constant multiple of the previous term. The simplest form of a geometric series is \( \sum_{n=0}^{\infty} ar^n \), where \( a \) is the first term and \( r \) is the common ratio. A crucial point about geometric series is the rule for convergence: a geometric series converges if the absolute value of the common ratio \( |r| < 1 \); otherwise, it diverges.The series \( \sum_{n=1}^{\infty} \frac{1}{2^n} \) is a classic example of a convergent geometric series, where \( a = \frac{1}{2} \) and \( r = \frac{1}{2} \). With a common ratio less than 1, this series sum converges to a finite number. In the context of comparing with the more complex series \( \sum_{n=1}^{\infty} \frac{1-n}{n2^n} \), it provides a benchmark. By demonstrating its convergence through known properties, it helps establish the likely convergence of other more complicated infinite series, offering a simpler path to determining behavior.