Problem 8 - Calculus Early Transcendentals: Pearson New International Edition

Question

In Problems 1-14, 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 Type

The given series is \( \sum_{k=1}^{\infty} \frac{3}{k} \). This is a harmonic series, modified with a constant factor 3. A harmonic series is of the form \( \sum_{k=1}^{\infty} \frac{1}{k} \). Since it resembles a harmonic series, we must investigate its behavior.

Step 2: Determine the Nature of the Series

Harmonic series are known to be divergent. The series \( \sum_{k=1}^{\infty} \frac{1}{k} \) diverges to infinity. Constant multiples of divergent series, such as \( \sum_{k=1}^{\infty} \frac{3}{k} \), also diverge.

Step 3: Evaluate the Convergence Using a Convergence Test

One common test for convergence is the \( p \)-series test. A series \( \sum_{k=1}^{\infty} \frac{1}{k^p} \) converges if \( p > 1 \). In our series, \( p = 1 \), which does not satisfy the condition for convergence, therefore the series diverges.

Key Concepts

Harmonic Series

The harmonic series is a fascinating and important topic in calculus. It refers to an infinite series that takes the form \( \sum_{k=1}^{\infty} \frac{1}{k} \). This series arises naturally in many mathematical contexts and is a classic example to demonstrate the concept of divergence.A simple way to think about the harmonic series is to imagine summing an infinite sequence of reciprocals of natural numbers:

• 1
• 1/2
• 1/3
• 1/4
• and so on...

Even though each term in the sequence becomes smaller and smaller, the sum of the series does not settle into a finite number. Instead, it grows without bound, illustrating the fascinating concept of divergence. A common modified version of this series includes multiplying the terms by a constant factor, such as \(\sum_{k=1}^{\infty} \frac{3}{k} \), which still maintains the divergent nature, but grows at a different rate.

p-Series Test

The \( p \)-series test is a handy tool in calculus to determine if a series converges. It helps us classify series of the form \( \sum_{k=1}^{\infty} \frac{1}{k^p} \). Understanding this test is crucial for analyzing various series where the exponent, \( p \), plays a pivotal role.

• If \( p > 1 \), the series will converge. This means the sum will approach a finite number as more terms are added.
• If \( p \leq 1 \), the series will diverge. In other words, the sum grows indefinitely.

For example, in the case of the harmonic series \( \sum_{k=1}^{\infty} \frac{1}{k} \), \( p = 1 \). Since the exponent \( p \) is equal to 1, the p-series test confirms that it diverges. This highlights the importance of the p-series test in determining the nature of a series.

Divergent Series

A divergent series is a crucial concept in calculus, referring to any series whose sum does not converge to a finite number. When dealing with series, one main goal is to ascertain whether it's convergent or divergent. This helps in understanding the behavior and limits of a series.Divergent series can take many forms, but a common characteristic is their tendency to grow without limit.Some key points about divergent series include:

• The sum can grow indefinitely, which is typically the case with harmonic series.
• A divergent series can also oscillate, where it does not settle towards any particular number or value.

In our exercise, the series \( \sum_{k=1}^{\infty} \frac{3}{k} \) serves as an example. Since it's closely related to the harmonic series, it demonstrates classic divergence due to the constant multiplier not affecting the harmonic series' tendency to grow without bound. Understanding why a series is divergent helps solidify your grasp of series convergence concepts.