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Magazine Chapter 8: Problem 18
Determine whether the following series converge. $$\sum_{k=2}^{\infty}(-1)^{k} \frac{\ln k}{k^{2}}$$
Short Answer
Expert verified
Answer: The series converges.
Step by step solution
01
Identify the terms of the alternating series
The given series is
$$\sum_{k=2}^{\infty}(-1)^{k} \frac{\ln k}{k^{2}}$$
The terms of the series have the form \((-1)^{k}b_k\), where \(b_k = \frac{\ln k}{k^{2}}\).
02
Determine if the sequence of absolute values is decreasing
We need to show that for all \(k \geq 2\), \(b_k \geq b_{k+1}\). Let's check the difference \(b_k - b_{k+1}\):
$$b_{k} - b_{k+1} = \frac{\ln k}{k^{2}} - \frac{\ln (k+1)}{(k+1)^2}$$
Now let's find its sign with inequalities.
Since \(k \geq 2\) and \(\ln\) is an increasing function, we have \(\ln k \leq \ln (k+1)\). Also, we have \((k+1)^2 \ge k^2\), so \(k^2 \le (k+1)^2\). Thus, we have:
$$0 \le \frac{\ln k}{(k+1)^2} \le \frac{\ln (k+1)}{(k+1)^2}$$
Then, it follows that \(0 \le b_k - b_{k+1} \le \frac{\ln k}{k^2} - \frac{\ln (k+1)}{k^2}\). As a result, we found that \(b_k \geq b_{k+1}\) for all \(k \geq 2\), so the sequence of absolute values is decreasing.
03
Determine if the sequence of absolute values converges to 0
We want to prove that \(b_k = \frac{\ln k}{k^2} \to 0\) as \(k \to \infty\). Since the logarithm function increases more slowly than any power function, it is true that \(\ln k = o(k)\) as \(k \to \infty\). Therefore, as k goes to infinity, \(b_k = \frac{o(k)}{k^2} = o(\frac{1}{k})\). Thus, \(b_k \to 0\).
By the Alternating Series Test, the given series converges since the sequence of absolute values is decreasing and converges to 0.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Convergence
Convergence is a vital concept in understanding whether a series adds up to a finite value. For a series to converge, the sum of its terms should approach a specific limit. In the context of the given series \(\sum_{k=2}^{\infty}(-1)^{k} \frac{\ln k}{k^{2}}\),\ \ the goal is to determine if it results in a finite sum.
To test for convergence, one common method is the Alternating Series Test, which relies on two main criteria:
Understanding convergence not only helps in grasping the behavior of infinite series but also in various fields like physics and engineering where such series occur frequently.
To test for convergence, one common method is the Alternating Series Test, which relies on two main criteria:
- The absolute values of the terms must form a decreasing sequence. This means each term is smaller than the previous one when ignoring their signs.
- The absolute values of the terms must approach zero as \(k\) increases. In simpler terms, as you keep adding more terms, the sizes of these terms must keep getting closer to zero.
Understanding convergence not only helps in grasping the behavior of infinite series but also in various fields like physics and engineering where such series occur frequently.
Absolute Values
Absolute values play a crucial role in determining the convergence of an alternating series. When we consider an alternating series, the terms alternate in sign, meaning they switch between positive and negative.
For example, the series \(\sum_{k=2}^{\infty}(-1)^{k} \frac{\ln k}{k^{2}}\),\ \ has terms such as \((-1)^{k}b_k\), where \(b_k = \frac{\ln k}{k^{2}}\). The absolute value of these terms is simply \(b_k = \frac{\ln k}{k^{2}}\), removing the alternation in sign.
In testing the convergence of an alternating series, it's crucial to look at the sequence formed by these absolute values, \(\{b_k\}\):
For example, the series \(\sum_{k=2}^{\infty}(-1)^{k} \frac{\ln k}{k^{2}}\),\ \ has terms such as \((-1)^{k}b_k\), where \(b_k = \frac{\ln k}{k^{2}}\). The absolute value of these terms is simply \(b_k = \frac{\ln k}{k^{2}}\), removing the alternation in sign.
In testing the convergence of an alternating series, it's crucial to look at the sequence formed by these absolute values, \(\{b_k\}\):
- We need to check if this sequence is decreasing. This ensures that as you go further in the series, each term has a diminishing impact.
- We also need to ensure \(b_k\) approaches zero. As the terms get smaller, they eventually stop impacting the overall sum of the series.
Alternating Series Test
The Alternating Series Test is a straightforward yet powerful tool to decide the convergence of a series with alternating signs. An alternating series is a sequence of terms that alternates in sign, going positive, negative, positive again, and so on.
This test involves examining two key conditions:
For the given series, both conditions are satisfied, meaning the series converges. The beauty of the Alternating Series Test lies in its simplicity. By just confirming these two properties, one can decisively conclude whether an alternating series adds up to a definite sum without needing to calculate the entire sum.
This critical test finds applications across mathematical problems and theoretical research, enhancing understanding of series behavior across various disciplines.
This test involves examining two key conditions:
- The series of absolute values, \(b_k = \frac{\ln k}{k^{2}}\), must be decreasing. This means that each successive term is smaller than the one before, which we verified earlier in the step-by-step solution.
- These absolute values must tend toward zero as \(k\) approaches infinity, indicating that the series diminishes in size, resulting in a controlled summation toward a finite value.
For the given series, both conditions are satisfied, meaning the series converges. The beauty of the Alternating Series Test lies in its simplicity. By just confirming these two properties, one can decisively conclude whether an alternating series adds up to a definite sum without needing to calculate the entire sum.
This critical test finds applications across mathematical problems and theoretical research, enhancing understanding of series behavior across various disciplines.
