Problem 29 - Calculus: Early Transcendentals

Question

Determine whether the series is convergent or divergent. If it is convergent, find its sum. \( \displaystyle \sum_{n = 1}^{\infty} \frac {2 + n}{1 - 2n} \)

Step-by-Step Solution

Answer

The series diverges.

Step 1: Identify the Series

We have the series \( \sum_{n=1}^{\infty} \frac{2+n}{1-2n} \). We need to analyze whether it converges or diverges. The general term of this series can be defined as \( a_n = \frac{2+n}{1-2n} \).

Step 2: Analyze the Terms

Notice that as \( n \) increases, both the numerator \( 2+n \) and the denominator \( 1-2n \) are linear functions. As \( n \to \infty \), the term \( 1-2n \) tends towards \(-\infty\), while \( 2+n \) tends towards \(\infty\). This means that \( a_n \) resembles the behavior of \(-\frac{n}{2n} \approx -\frac{1}{2} \) for large \( n \).

Step 3: Determine Divergence or Convergence

Given \( a_n \approx -\frac{n}{2n} = -\frac{1}{2} \), the terms of the series do not tend to zero, as a condition for convergence of a series is that \( \lim_{n \to \infty} a_n = 0 \). Since this condition is not satisfied (specifically, \( \lim_{n \to \infty} a_n = -\frac{1}{2} \)), the series diverges.

Key Concepts

Convergent Series

A series is considered convergent if its terms approach zero as they progress to infinity, and the sum of the series approaches a finite number. In simple terms, it means that as we keep adding more terms, the total sum gets closer and closer to a specific value.
To determine if a series converges, we analyze the series' terms. Specifically, we're interested in whether the limit of the sequence of terms goes to zero as the sequence extends to infinity. Mathematically, a series \( \sum_{n=1}^{\infty} a_n \) is convergent if:

• The limit \( \lim_{n \to \infty} a_n = 0 \)
• The sequence of partial sums \( S_n \) of the series is convergent

If these conditions are met, the series converges.
An example of a convergent series is the geometric series with a common ratio \(r\) where \(|r| < 1\). Here, the series \( \sum_{n=0}^{\infty} ar^n \) converges to \( \frac{a}{1-r} \).
It's essential to understand that the concept of convergence helps in identifying series where adding more terms will not disrupt the final sum that the series approaches.

Divergent Series

In contrast to convergent series, divergent series are those that do not meet the criteria for convergence. That means the sum of their terms does not tend toward a finite number as more terms are added.
A series \( \sum_{n=1}^{\infty} a_n \) is considered divergent if:

• The limit \( \lim_{n \to \infty} a_n eq 0 \)
• The sequence of partial sums does not converge to a finite value

One key point to remember is that if the limit of the terms of a series doesn't go to zero, the series cannot converge. Therefore, it must diverge.
In our original exercise, the series \( \sum_{n=1}^{\infty} \frac{2+n}{1-2n} \) demonstrates divergent behavior because the limit of its terms is \(-\frac{1}{2}\), not zero. As the terms do not diminish sufficiently to zero, the series expands infinitely without settling towards a finite sum.

Infinite Series Analysis

Infinite series analysis involves studying the behavior of an endless succession of terms. Understanding the convergence or divergence of these series is crucial in various applications, such as mathematical modeling, physics, and engineering.
There are several techniques we employ in infinite series analysis to decide convergence or divergence.

• Limit comparison test: Compare the series with a known benchmark series.
• Ratio test: Consider the limit of the ratio of consecutive terms.
• Root test: Evaluate the \(n\)-th root of the absolute value of terms.
• Alternating series test: For series with alternating positive and negative terms.

By using these methods, we can rigorously determine whether series converge or diverge. Our original series, from the exercise, fell under general analysis where term limits helped confirm its divergence.
A solid grasp of infinite series evaluation allows deeper insights into complex mathematical relationships, making it easier to find solutions to significant problems in other scientific fields.