Problem 18 - Thomas Calculus

Question

Which of the series in Exercises \(11-40\) converge, and which diverge? Give reasons for your answers. (When you check an answer, remember that there may be more than one way to determine the series' convergence or divergence.) $$ \sum_{n=1}^{\infty} \frac{-8}{n} $$

Step-by-Step Solution

Answer

The series diverges.

Step 1: Identify the Type of Series

The given series is \( \sum_{n=1}^{\infty} \frac{-8}{n} \). Notice that it takes the form of a general Harmonic series \( \sum_{n=1}^{\infty} \frac{c}{n} \), where \( c \) is a constant. In this case, \( c = -8 \).

Step 2: Determine the Convergence or Divergence

A harmonic series \( \sum_{n=1}^{\infty} \frac{c}{n} \) diverges if \( c eq 0 \). This is because the series \( \sum_{n=1}^{\infty} \frac{1}{n} \) itself is known as the harmonic series and is divergent. Therefore, the series \( \sum_{n=1}^{\infty} \frac{-8}{n} \) also diverges.

Step 3: Use the Integral Test (Optional)

To further verify divergence, consider the integral test: \( \int_{1}^{\infty} \frac{-8}{x} \, dx \). This integral evaluates to \( \lim_{t \to \infty} [-8 \ln x]_{1}^{t} = -8(\ln t - \ln 1) = -8 \ln t \). As \( t \to \infty \), \( \ln t \to \infty \), thus the integral diverges, confirming our earlier finding.

Key Concepts

Harmonic Series

A harmonic series is a fundamental type of infinite series, typically expressed as \( \sum_{n=1}^{\infty} \frac{1}{n} \). This series stands as a notable example in mathematical analysis because it is an example of a divergent series. Despite the terms \( \frac{1}{n} \) decreasing as \( n \) grows, the harmonic series does not converge to a finite number. Essentially, as you keep adding the terms, the sum continues to grow indefinitely.

• It is a classic example of how a series may appear to be getting smaller, but the sum continues to grow without bound.
• For any constant \( c eq 0 \), the series \( \sum_{n=1}^{\infty} \frac{c}{n} \) will likewise diverge, as multiplying by a constant does not change the divergent nature of the harmonic series.

When analyzing series like \( \sum_{n=1}^{\infty} \frac{-8}{n} \), understanding the harmonic series helps recognize their behavior as divergent.

Integral Test

The integral test is a valuable tool in determining the convergence or divergence of infinite series. It connects the sum of a series to a related improper integral to make the determination. The essence of the test is:

• If \( f(x) \) is a continuous, positive, decreasing function for \( x \ge a \) and \( f(n) = a_n \), then the series \( \sum_{n=a}^{\infty} a_n \) converges if and only if the integral \( \int_{a}^{\infty} f(x) \, dx \) converges.

Using the integral test on the series \( \sum_{n=1}^{\infty} \frac{-8}{n} \), we can evaluate the integral \( \int_{1}^{\infty} \frac{-8}{x} \, dx \). This integral gives us \( \lim_{t \to \infty} [-8 \ln x]_{1}^{t} \). As \( t \to \infty \), \( \ln t \to \infty \), leading to divergence.
Thus, the integral test confirms that a series of this form diverges.

Divergent Series

In the world of series, a divergent series is one whose terms do not approach a specific limit. It means that instead of creeping closer to a fixed sum, the series either grows indefinitely or oscillates without settling down.

• A divergent series does not have a sum in the conventional sense.
• The harmonic series, \( \sum_{n=1}^{\infty} \frac{1}{n} \), is a principal example of this type.

When considering a specific divergent series, such as \( \sum_{n=1}^{\infty} \frac{-8}{n} \), we recognize this as simply a variation governed by the harmonic progression. Because the negative constant has no impact on the divergent nature of the harmonic series, this series diverges as well. It's critical to first recognize the nature of a series before attempting to analyze if it's convergent or divergent.