Problem 8 - Calculus Early Transcendentals: Pearson New International Edition

Question

Indicate whether the given series converges or diverges. If it converges, find its sum. Hint: It may help you to write out the first few terms of the series $$ \sum_{k=1}^{\infty} \frac{3}{k} $$

Step-by-Step Solution

Answer

The series diverges.

Step 1: Identify the series

The series given is \( \sum_{k=1}^{\infty} \frac{3}{k} \). This is an infinite series where each term is of the form \( \frac{3}{k} \), which is similar to the harmonic series \( \sum_{k=1}^{\infty} \frac{1}{k} \) but multiplied by a constant 3.

Step 2: Recognize the type of series

The series \( \sum_{k=1}^{\infty} \frac{3}{k} \) is a type of p-series, specifically the harmonic series multiplied by a constant. The harmonic series is known to diverge.

Step 3: Apply Divergence Test

The Divergence Test states if the limit of the sequence terms \( a_k \) as \( k \to \infty \) is not zero, the series \( \sum_{k=1}^{\infty} a_k \) diverges. For \( \frac{3}{k} \), the limit approaches zero as \( k \to \infty \), so the divergence test is inconclusive.

Step 4: Conclude Divergence based on known results

Since the terms resemble the harmonic series, and it is known that \( \sum_{k=1}^{\infty} \frac{1}{k} \) diverges, multiplying by a constant does not affect the divergence. Therefore, the series \( \sum_{k=1}^{\infty} \frac{3}{k} \) must also diverge.

Key Concepts

Harmonic Series

The harmonic series is one of the most famous infinite series in mathematics. It is defined as:

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

Here, each term takes the form of the reciprocal of a positive integer.

The harmonic series is well-known because it is one of the simplest examples of a series that diverges. Diverges means that as you keep adding more and more terms, the sum grows without bound.

Understanding the harmonic series is crucial because it sets the stage for comprehending more complex series. Although each term gets smaller and smaller, the series as a whole does not settle to a fixed number.

Divergence Test

The divergence test, also known as the nth-term test, is a simple and fundamental method used to determine if a series diverges.

The test says:

• If \( \lim_{k \to \infty} a_k eq 0 \), then the series \( \sum_{k=1}^{\infty} a_k \) diverges.

However, what if the limit is zero? In such a case, the test cannot conclude whether the series converges or diverges. It's a necessary condition for convergence, but not sufficient.

In the problem given, despite each term \( \frac{3}{k} \) approaching zero, the series still diverges. This emphasizes the limitation of the divergence test.

Infinite Series

An infinite series is essentially a sum with an infinite number of terms. These series can either converge to a fixed number or diverge as they grow indefinitely.

Key aspects of infinite series include:

• Convergence: The series settles to a specific value.
• Divergence: The series continues to grow without approaching a specific limit.

Recognizing whether a series converges or diverges is crucial in understanding its behavior. For instance, in the exercise, the series \( \sum_{k=1}^{\infty} \frac{3}{k} \) does not converge, suggesting it behaves similarly to an ever-growing sum.

P-Series

The term "p-series" refers to a mathematical series of the form:

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

In these series, \( p \) is a positive real number that determines convergence.

The rules of convergence for a p-series are:

• If \( p > 1 \), the series converges.
• If \( p \leq 1 \), the series diverges.

For example, the harmonic series has \( p = 1 \), and hence it diverges.

By recognizing the given series as a p-series with \( p = 1 \) multiplied by a constant, it's clear that it must diverge just like the harmonic series itself.