Problem 44 - Thomas Calculus

Question

Are there any values of \(x\) for which \(\sum_{n=1}^{\infty}(1 / n x)\) converges? Give reasons for your answer.

Step-by-Step Solution

Answer

No values of \( x \) make the series converge.

Step 1: Understanding the Problem

We are asked to find if there are any values of \( x \) for which the infinite series \( \sum_{n=1}^{\infty}\frac{1}{nx} \) converges. This is a series where each term is \( \frac{1}{nx} \).

Step 2: Identifying the Form of the Series

The series given by \( \sum_{n=1}^{\infty}\frac{1}{nx} \) is a harmonic series multiplied by \( \frac{1}{x} \). A harmonic series \( \sum_{n=1}^{\infty}\frac{1}{n} \) is known to diverge.

Step 3: Considering the Impact of \( x \)

If we multiply a divergent series by a constant \( \frac{1}{x} \) (where \( x eq 0 \)), the series remains divergent because the behavior of the series pertaining to convergence does not change by multiplying each term by the same non-zero constant.

Step 4: Special Case for \( x = 0 \)

If \( x = 0 \), each term in the series becomes undefined as division by zero is not permissible. Therefore, \( x = 0 \) is not a valid consideration for convergence.

Step 5: Concluding the Analysis

Since the series is divergent for any non-zero \( x \) and undefined for \( x = 0 \), there are no values of \( x \) for which \( \sum_{n=1}^{\infty}\frac{1}{nx} \) converges.

Key Concepts

Harmonic Series

The harmonic series is a classic example in mathematics that provides great insight into understanding convergence and divergence of series. A harmonic series has the form:

• \( \sum_{n=1}^{\infty} \frac{1}{n} \)

This series is made of terms that decrease as \(n\) increases, but it is famously divergent. Despite the terms becoming infinitely small, they do not sum up to a finite limit.

The divergence of the harmonic series might initially seem counterintuitive because individual terms approach zero. However, the series' rate of decreasing is not fast enough. Thus, when handling the harmonic series, keep in mind that it serves as a foundational example of a simple yet divergent series.

Infinite Series

Infinite series play an essential role in mathematics, allowing the sum of innumerable terms to be considered as a whole. When we talk about an infinite series, we indeed consider sequences that extend indefinitely, such as:

• \( \sum_{n=1}^{\infty} a_n \)

Here, each \(a_n\) represents a term in the series. Whether an infinite series converges or diverges can greatly affect its application in mathematics, physics, and engineering.

A series is considered convergent if the sum of its terms approaches a specific value as more terms are added. Conversely, if the sum continues to grow without bounds, it is classified as divergent. The concept of convergence is crucial because only convergent series can be assigned a meaningful value that represents their 'sum' even though they consist of infinite terms.

Convergence and Divergence

The concepts of convergence and divergence determine the behavior of an infinite series. To check if a series converges, one often applies various tests like the Comparison Test or Ratio Test. If no such test concludes convergence definitively, the series is usually divergent.

When a series converges, it means:

• The sum of its terms approaches a specific finite number.
• The terms \(a_n\) decrease rapidly enough so that adding them all up leads to a finite result.

In contrast, a diverging series behaves differently. It may either:

• Continue growing indefinitely large.
• Not settle into a single value, potentially oscillating as terms are added.

Understanding these properties is important for solving mathematical problems, such as determining the potential values of \(x\) for which a given series converges. As shown in the problem, multiplying a known divergent series like the harmonic series by any constant other than zero, keeps it divergent. This happens because such operations do not alter the essential characteristics leading to divergence.