Problem 41 - 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{2^{n}-n}{n 2^{n}}\end{equation}

Step-by-Step Solution

Answer

The series diverges because the harmonic component outweighs the convergent geometric component.

Step 1: Identify the Series Formula

The given series is \(\sum_{n=1}^{\infty} \frac{2^{n}-n}{n 2^{n}}\). To understand if it converges or diverges, we need to analyze the general term of the series \(a_n = \frac{2^n - n}{n 2^n}\).

Step 2: Simplify the General Term

Let's simplify the general term:\[a_n = \frac{2^n - n}{n 2^n} = \frac{2^n}{n 2^n} - \frac{n}{n 2^n} = \frac{1}{n} - \frac{1}{2^n}\]. The general term consists of two parts: \(\frac{1}{n}\) and \(\frac{1}{2^n}\).

Step 3: Analyze Each Component Separately

First, analyze \(\sum \frac{1}{n}\): This represents the harmonic series, which is known to diverge.Next, analyze \(\sum \frac{1}{2^n}\): This is a geometric series with a common ratio \(r = \frac{1}{2}\), where \(|r| < 1\), thus it converges.

Step 4: Determine Behavior of Combined Series

The original series \(\sum \left( \frac{1}{n} - \frac{1}{2^n} \right)\) can be viewed as a difference between a divergent series (harmonic series) and a convergent series (geometric series). In such a case, the larger divergent behavior dominates.

Step 5: Conclude the Convergence or Divergence

Since the divergent part, the harmonic series \(\sum \frac{1}{n}\), dictates the overall behavior of the series, the entire series \(\sum \left( \frac{1}{n} - \frac{1}{2^n} \right)\) diverges as well.

Key Concepts

Harmonic Series

The harmonic series is one of the most well-known examples of a divergent series. It is expressed as \[ \sum_{n=1}^{\infty} \frac{1}{n} = 1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \ldots \]At first glance, it might seem that since the terms get smaller, the series may converge. However, the harmonic series diverges. This means that as you keep adding more and more terms, the total continues to increase indefinitely without approaching a fixed limit.

Here's a simple way to understand why the harmonic series diverges:

• Group terms in pairs, triplets, or specific patterns to compare their sum, which shows how they exceed a particular benchmark.
• The sum grows continuously though at a slower pace, due to the ever-decreasing fraction values.

This property is central to analysis because it provides examples of how specific infinite series can behave despite smaller term values.

Geometric Series

The geometric series is a series with a constant ratio between consecutive terms. It is written as:\[ \sum_{n=0}^{\infty} ar^n = a + ar + ar^2 + ar^3 + \ldots \]where \(a\) is the first term, and \(r\) is the common ratio.

A geometric series can either converge or diverge, depending on the absolute value of the common ratio \(r\):

• If \(|r| < 1\), the series converges to \( \frac{a}{1-r} \).
• If \(|r| \geq 1\), the series diverges.

In our initial problem, the part \(\sum \frac{1}{2^n}\) forms a geometric series with \(a = 1\) and \(r = \frac{1}{2}\). Since the common ratio \(|r| = \frac{1}{2}\) is less than 1, this component converges, yielding a finite sum. Understanding geometric series is key in calculus as it assists in solving various problems involving exponential functions and decay processes.

Divergence

Divergence in series indicates that the sum of the series does not settle towards a fixed number as more terms are added. In simpler terms, instead of approaching a particular number, the series gets indefinitely larger or oscillates indefinitely.

In examining series, one often uses the divergence test: if the limit of the sequence of term values does not approach zero, the series must diverge. But even if the limit of the terms does approach zero, the series can still diverge, as seen in the harmonic series.

• For instance, in our original exercise, the overall series diverges because the divergent harmonic series overtakes the convergent geometric series.
• A critical point in determining behavior often comes from recognizing dominant properties in the series components.

Grasping divergence is crucial when working through series or understanding function behaviors, particularly in applied mathematics and engineering fields.