Problem 14 - 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 $$ \sum_{k=6}^{\infty} \frac{2}{k-5} $$

Step-by-Step Solution

Answer

The series diverges.

Step 1: Identify Series Type

The given series is \( \sum_{k=6}^{\infty} \frac{2}{k-5} \). Notice that this looks like a harmonic series but shifted starting from \( k-6 \). To clarify, write a few initial terms: for \( k=6 \), \( \frac{2}{6-5}=2 \); for \( k=7 \), \( \frac{2}{7-5}=1 \); for \( k=8 \), \( \frac{2}{8-5}=\frac{2}{3} \). This is a divergent harmonic series with a constant multiplication of 2.

Step 2: Comparison with Harmonic Series

The series can be rewritten as \( 2 \times \sum_{n=1}^{\infty} \frac{1}{n} \) after substituting \( n = k - 5 \). The harmonic series \( \sum_{n=1}^{\infty} \frac{1}{n} \) is known to diverge. Since we are multiplying by a constant, the divergence remains the same.

Step 3: Apply Divergence Test

According to the divergence test, if the terms of a series do not tend to zero, the series diverges. Here, for each term \( \frac{2}{k-5} \), it approaches zero as \( k \) tends towards infinity, but since the series is effectively the harmonic series, as seen from previous reasoning, we can confirm the series diverges by comparison.

Step 4: Conclusion

The given series \( \sum_{k=6}^{\infty} \frac{2}{k-5} \) diverges because it is equivalent to a divergent harmonic series multiplied by a constant factor.

Key Concepts

Divergence Test

The divergence test is a simple and effective tool to determine whether a series diverges.
It comes into play with series like the one in the exercise, where we need to establish if the series diverges right at the beginning.

To apply the divergence test, you essentially look at the individual terms of the series. If the limit of the terms does not approach zero as the series progresses towards infinity, then the series must diverge.

This makes logical sense; if the terms of a series do not vanish to zero, the series cannot reasonably settle on a finite sum.
Even when terms do approach zero, this test alone isn't a guarantee of convergence; it's merely a quick way to rule out any possible convergence when they fail to vanish.

• Check if \(\lim_{{k \to \infty}} a_k = 0\).
• If not, series diverges.
• If yes, further tests needed.

In the exercise example, while the terms do tend to zero, the series diverges—not ruled out solely by the divergence test, but using further analysis of its harmonic nature.

Harmonic Series

The harmonic series is a fundamental concept when studying series convergence. It's represented by \( \sum_{n=1}^{\infty} \frac{1}{n} \).
The series is fascinating because it diverges despite the terms tending towards zero.

In easier terms, as we go further along the series, the terms get smaller—but not small enough quickly enough to sum up to a finite number.
This characteristic leads to the harmonic series being known famously for its divergence.

• Terms: \( \frac{1}{1}, \frac{1}{2}, \frac{1}{3}, \ldots \)
• Nature: Diverges uniquely because terms shrink too slowly.

The exercise leverages this property by recognizing its series as a harmonic series potentially masked by additional constants—resulting in the same kind of divergence.

Infinite Series

An infinite series is, at its core, the sum of infinitely many terms. These series can be bewildering because they potentially carry on forever, but can still sometimes add up to a finite value.

The fundamental concern with infinite series is understanding when they converge to a sum.The most straightforward of infinite series are geometric and arithmetic series, but many other forms exist, like harmonic series as mentioned above.

• Composition: Involves endless terms.
• Behavior: Can converge or diverge.
• Goal: Identify overall trends, whether finite sum exists.

In this exercise, examining the series from \( k=6 \to \infty \) as an infinite series helps frame its divergence by recognizing underlying properties tied to harmonic behavior.

Comparison Test

The comparison test offers a methodical way to evaluate series by comparing it to another reference series whose behavior we understand better.

In application, you compare the series’ terms against terms of another known series—usually one with comparable growth rates or structures—and decipher its convergence or divergence accordingly.

• Purpose: Determines match or divergence based on familiar series.
• Usage: Find another series \( b_n \) with clear divergence/convergence.
• Comparison: Use \( 0 \leq a_n \leq b_n \) or \( a_n \geq b_n \geq 0 \) logic.

In the exercise, by reformulating terms and directly matching them against the harmonic series, we quickly establish that this series mirrors harmonic behavior and diverges similarly.
It's an effective shortcut, bypassing longer verbal reasoning by leveraging prior established knowledge.