Series Convergence: Finding 'p' For Absolute Convergence

Hey math enthusiasts! Let's dive into the fascinating world of series convergence. Today, we're tackling a classic problem: finding the values of p for which certain series converge absolutely. This is a crucial concept in calculus and analysis, so pay close attention. We'll break down each series step-by-step, making sure you grasp the underlying principles. Get ready to flex those mathematical muscles!

Part A: Unveiling the Convergence of { rac{1}{k(\log(k))^p} }

Alright, guys, let's kick things off with the series ${ \sum_{k=2}^{\infty} \frac{1}{k(\log(k))^p} }$. Our mission? Determine the values of p for which this series converges absolutely. This one's a prime candidate for the Integral Test. Remember that the Integral Test is a powerful tool for determining the convergence or divergence of an infinite series by comparing it to an improper integral. It's particularly useful when dealing with terms that can be easily integrated. For this specific series, the Integral Test is our best bet.

So, what's the deal with the Integral Test? If ${ f(x) }$ is a continuous, positive, and decreasing function on the interval ${ [a, \infty) }$, then the series ${ \sum_{k=a}^{\infty} f(k) }$ converges if and only if the integral ${ \int_{a}^{\infty} f(x) dx }$ converges. Let's get down to business. We'll consider the function ${ f(x) = \frac{1}{x(\log(x))^p} }$. This function is continuous, positive, and decreasing for ${ x \geq 2 }$. Now, let's set up the integral:

${\int_{2}^{\infty} \frac{1}{x(\log(x))^p} dx}$

To evaluate this integral, we'll use a u-substitution. Let ${ u = \log(x) }$, so ${ du = \frac{1}{x} dx }$. The limits of integration change as well: when ${ x = 2 }$, ${ u = \log(2) }$, and as ${ x \to \infty }$, ${ u \to \infty }$. Thus, our integral becomes:

${\int_{\log(2)}^{\infty} \frac{1}{u^p} du}$

Now, this is a much simpler integral to deal with. Recall that the integral ${ \int_{a}^{\infty} \frac{1}{u^p} du }$ converges if ${ p > 1 }$ and diverges if ${ p \leq 1 }$. Applying this to our situation, we find that the integral ${ \int_{\log(2)}^{\infty} \frac{1}{u^p} du }$ converges if ${ p > 1 }$ and diverges if ${ p \leq 1 }$. Since the convergence of the integral is equivalent to the convergence of the series (by the Integral Test), we conclude that the series ${ \sum_{k=2}^{\infty} \frac{1}{k(\log(k))^p} }$ converges absolutely when p > 1. If p is less than or equal to 1, then the series diverges. Pretty neat, huh?

So, to recap, for the series ${ \sum_{k=2}^{\infty} \frac{1}{k(\log(k))^p} }$ to converge absolutely, the value of p must be greater than 1. This result highlights the critical role of p in determining the convergence behavior of the series. Understanding this helps us predict and analyze the behavior of various mathematical models and systems. The application of the Integral Test here underscores its importance in series analysis.

Now you might be wondering, what happens if p equals 1? Well, the series becomes ${ \sum_{k=2}^{\infty} \frac{1}{k\log(k)} }$, and we know that it diverges. This is a classic example that you should probably remember. Keep in mind that the Integral Test is not just about getting an answer; it’s about understanding the relationship between series and integrals.

Keep practicing, and you'll become a convergence guru in no time!

Part B: Exploring the Convergence of ${ \sum_{k=2}^{\infty} \frac{1}{\cos(k)^5} }$

Okay, let's switch gears and tackle the series ${ \sum_{k=2}^{\infty} \frac{1}{\cos(k)^5} }$. This one looks a bit trickier because of the cosine function. Remember, the cosine function oscillates between -1 and 1. This oscillation is a key element in understanding the behavior of the series. We must be very careful when dealing with trigonometric functions in series, as they can sometimes lead to unexpected results.

The core issue here is that ${ \cos(k) }$ does not approach zero as k goes to infinity. Furthermore, the values of ${ \cos(k) }$ are not uniformly bounded away from zero. Therefore, we can't directly apply the tests we used in Part A. Notice that the terms of the series, ${ \frac{1}{\cos(k)^5} }$, do not approach zero as k increases. This is a critical observation because, for a series to converge, its terms must approach zero. This is a necessary (but not sufficient) condition for convergence. The fact that the terms don't go to zero immediately tells us that the series diverges.

Think about it: the denominator, ${ \cos(k)^5 }$, will take on values close to 1 and -1, as well as values in between. This means the fraction will oscillate and not settle down to a single value. This oscillating behavior prevents the series from converging. In other words, the terms of the series do not get arbitrarily small. Thus, based on the divergence test, we can immediately conclude that the series diverges.

More specifically, the divergence test states that if the limit of the terms of a series does not equal zero, then the series diverges. In our case, ${ \lim_{k\to\infty} \frac{1}{\cos(k)^5} }$ does not exist (and certainly doesn't equal zero) because of the oscillatory nature of the cosine function. This is why we can say, with confidence, that this series diverges.

To be perfectly clear, this series does not converge absolutely or conditionally. The divergence test provides us with a quick and effective way to determine this without having to dive into more complex convergence tests. The take-away here is to always check the basic requirements of convergence, such as the limit of the terms approaching zero, before applying more advanced tests.

So, to sum up, the series ${ \sum_{k=2}^{\infty} \frac{1}{\cos(k)^5} }$ diverges for all values of p. No matter what p is, this series will not converge because the terms do not approach zero. This is a critical insight, guys!

Part C: Analyzing the Series ${ \sum_{k=1}^{\infty} \dots }$

Okay, it looks like there's a problem with the prompt because we don't have the full series to analyze for Part C. However, let's explore some general strategies for series convergence analysis. This will prepare you for whatever series comes your way!

When you're faced with a series, start by carefully examining the terms. What do they look like? Do they involve factorials, exponentials, or trigonometric functions? The form of the terms often guides you toward the appropriate convergence test.

Here's a quick rundown of some useful tests:

• The Divergence Test: Always the first check! Does the limit of the terms equal zero? If not, the series diverges.
• The Ratio Test: Great for series with factorials or exponentials. It compares the ratio of consecutive terms. If the limit of the ratio is less than 1, the series converges absolutely.
• The Root Test: Useful when the terms involve powers. It takes the nth root of the absolute value of the terms and examines the limit.
• The Integral Test: (As we saw in Part A) Works well when the terms can be easily integrated.
• The Comparison Test and Limit Comparison Test: These tests compare the given series to a series whose convergence or divergence is already known. They are helpful for series that are similar to known series.
• Alternating Series Test: Applies to series with alternating signs. It requires the terms to decrease in magnitude and approach zero.

Now, let's talk about absolute convergence. A series ${ \sum a_k }$ converges absolutely if the series ${ \sum |a_k| }$ converges. If a series converges absolutely, then it also converges. However, the converse is not always true; a series can converge conditionally (i.e., converge but not absolutely).

When we have the complete series for Part C, we'd need to first determine the general form of the terms. Then we would apply the appropriate test (or tests) to determine the values of p for which it converges absolutely. For this analysis, knowing the exact series terms is crucial. Based on the terms we could then determine how to tackle it, whether through the Ratio Test, Root Test, Integral Test, or others.

In the meantime, keep practicing and studying the different convergence tests. Understanding these tests is the key to mastering series convergence! Also, remember that practice makes perfect, so work through many different examples to solidify your understanding.

So there you have it, a thorough discussion on series convergence! Keep up the great work, and happy calculating!