Problem 2 - 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 $$ \sum_{k=1}^{\infty}\left(-\frac{1}{4}\right)^{-k-2} $$

Step-by-Step Solution

Answer

The series converges, and its sum is \(-\frac{1}{80}\).

Step 1: Rewrite the Series

The given series is \( \sum_{k=1}^{\infty}\left(-\frac{1}{4}\right)^{-k-2} \). We need to manipulate the expression of each term to better understand the series. Each term can be rewritten as \( \left(-\frac{1}{4}\right)^{-k-2} = \left(\frac{1}{-4}\right)^{k+2} \). This simplifies further to \( \left(-\frac{1}{4}\right)^{k+2}\). Thus, our series becomes a geometric series: \( \sum_{k=1}^{\infty}\left(-\frac{1}{4}\right)^{k+2} \).

Step 2: Identify the Geometric Series Parameters

From the rewritten series \( \sum_{k=1}^{\infty}\left(-\frac{1}{4}\right)^{k+2} \), we can identify that the first term (\( a \)) when \( k = 1 \) is \( \left(-\frac{1}{4}\right)^3 = -\frac{1}{64} \) and the common ratio (\( r \)) is \( -\frac{1}{4} \).

Step 3: Determine Convergence

A geometric series \( \sum_{k=0}^\infty ar^k \) converges if and only if \( |r| < 1 \). In this case, the common ratio \( r = -\frac{1}{4} \) meets the convergence criteria since \( |-\frac{1}{4}| = \frac{1}{4} < 1 \). Therefore, the series is convergent.

Step 4: Sum the Convergent Series

For a convergent geometric series, the sum can be found using the formula \( S = \frac{a}{1-r} \), where \( a \) is the first term and \( r \) is the common ratio. Here, \( a = -\frac{1}{64} \) and \( r = -\frac{1}{4} \). Substituting these values gives:\[S = \frac{-\frac{1}{64}}{1 - (-\frac{1}{4})} = \frac{-\frac{1}{64}}{1 + \frac{1}{4}} = \frac{-\frac{1}{64}}{\frac{5}{4}} = -\frac{1}{64} \times \frac{4}{5} = -\frac{1}{80}\]

Step 5: Conclusion

The series \( \sum_{k=1}^{\infty}\left(-\frac{1}{4}\right)^{-k-2} \) converges, and its sum is \(-\frac{1}{80}\).

Key Concepts

Series Convergence

In the world of infinite series, determining whether a series converges or diverges is a fundamental topic. Convergence means that as you keep adding the terms of the series, the total sum approaches a specific value. For divergence, the sum might increase indefinitely or behave erratically without settling down. This behavior is essential, as convergent series can be manipulated, summed, and used in mathematical applications more reliably.

To understand convergence, imagine you're filling up a container with more and more water, yet it never spills over. The infinite series keeps adding terms, but it settles into a specific value that doesn't change as more terms are added. This concept is different from divergent series, where there's no limit in sight, and the total keeps changing drastically.

When dealing with infinite series, one of the most reliable ways to test convergence is through specific convergence criteria that fit the nature of the series, such as the size of the terms or relationships between them.

Geometric Sequence

A geometric sequence is a special kind of sequence in mathematics where each term after the first is found by multiplying the previous one by a constant, called the common ratio (\( r \)). For instance, in a sequence like 2, 4, 8, 16, each number is obtained by multiplying the previous one by 2, thus 2 is the common ratio.

When we extend this concept to an infinite series, we can have a geometric series. In our given problem, the series involves terms that can be represented as powers of \( \left(-\frac{1}{4}\right) \). By recognizing this series as a geometric series, it becomes easier to analyze it and determine its behavior, like its convergence.

Every geometric series has its own specific first term and common ratio. These elements serve as the foundation for further analysis, like finding the sum if the series is convergent. Understanding these fundamental properties helps in expanding our capacity to work with many types of sequences and series.

Convergence Criteria

The convergence criteria for geometric series play a critical role in establishing whether a series will settle at a particular sum or not. For a geometric series to converge, the absolute value of its common ratio (\( r \)) must be less than 1, symbolically represented as \( |r| < 1 \).

This condition ensures that each successive term gets smaller and smaller, allowing the series to "home in" on a specific sum. For example, in the given series, \( |r| = \left|-\frac{1}{4}\right| = \frac{1}{4} < 1 \), which meets the criteria for convergence.

Understanding this criterion is not only crucial for solving problems about series convergence but also aids in the comprehension of concepts like limits and asymptotic behavior in mathematical studies. It forms the backbone of more advanced mathematical theories and applications, allowing us to predict and manipulate the outcomes of infinite series with precision.