Series Convergence: Does It Converge Or Diverge?

Nov 2, 2025 by Admin 49 views

$Does the series shown below converge or diverge? $\sum_{n=2}^{\infty} \frac{\ln (n+5)}{n+5}$$

Hey math enthusiasts! Let's dive into the fascinating world of infinite series and figure out whether the series $\sum_{n=2}^{\infty} \frac{\ln (n+5)}{n+5}$ converges or diverges. This is a classic problem in calculus, and understanding the behavior of infinite series is super important for many areas of math and science. So, grab your pencils and let's get started! We'll break down the problem step by step to make it as clear as possible.

Understanding Convergence and Divergence

Before we jump into the specific series, let's refresh our memory on what it means for a series to converge or diverge. Imagine you have an infinite sum of numbers.

If the sum approaches a finite value as you keep adding more terms, the series converges. Think of it like a journey that eventually reaches a destination.
If the sum doesn't approach a finite value, the series diverges. This could mean the sum goes off to infinity or oscillates without settling down. It's like a never-ending road with no final destination.

Our goal is to figure out which of these two scenarios applies to our series. There are several tests we can use to determine the convergence or divergence of a series. One effective method for this particular series is the Integral Test. Let's see how it works!

Applying the Integral Test

The Integral Test is a powerful tool, guys, especially when dealing with series involving functions that are easy to integrate. The test says that if we have a series of the form $\sum_{n=1}^{\infty} f(n)$ , where f(x) is a continuous, positive, and decreasing function for x ≥ 1, then the series converges if and only if the integral $\int_{1}^{\infty} f(x) dx$ converges.

Here's how we can apply it to our series $\sum_{n=2}^{\infty} \frac{\ln (n+5)}{n+5}$ :

Define the function: First, we replace n with x to create a continuous function: $f(x) = \frac{\ln (x+5)}{x+5}$.
Check the conditions: We need to make sure f(x) is continuous, positive, and decreasing for x ≥ 2.
- Continuous: The function is continuous for x > -5, so it's definitely continuous for x ≥ 2.
- Positive: Since x + 5 > 0 for x ≥ 2, and ln(x + 5) is positive for x > -4, f(x) is positive for x ≥ 2.
- Decreasing: To check if f(x) is decreasing, we can look at its derivative. If the derivative is negative, the function is decreasing. Let's find the derivative of f(x) using the quotient rule: $f'(x) = \frac{(\frac{1}{x+5})(x+5) - \ln(x+5)}{(x+5)^2} = \frac{1 - \ln(x+5)}{(x+5)^2}$. For *x* > *e* - 5 ≈ -1.7, the natural log will be greater than 1, and the derivative will be negative. This means *f(x)* is decreasing for *x* > *e* - 5. And since our series starts at *n* = 2, the function is definitely decreasing in the relevant range.$
Evaluate the integral: Now, let's evaluate the improper integral: $\int_{2}^{\infty} \frac{\ln (x+5)}{x+5} dx$.

To do this, we'll use a substitution. Let u = ln(x + 5). Then, du = (1/(x + 5)) dx. The limits of integration will also change. When x = 2, u = ln(7). As x approaches infinity, u also approaches infinity. So, the integral becomes: $\int_\ln(7)}^{\infty} u du$. Integrating, we get $\left[ \frac{1{2}u^2 \right]_{\ln(7)}^{\infty}$. As u goes to infinity, u² also goes to infinity. Therefore, the integral diverges.

Conclusion: Divergence

Since the integral $\int_{2}^{\infty} \frac{\ln (x+5)}{x+5} dx$ diverges, according to the Integral Test, the series $\sum_{n=2}^{\infty} \frac{\ln (n+5)}{n+5}$ diverges as well. This means that the sum of the series does not approach a finite value; instead, it grows without bound. We can confidently say that the series does not converge, and therefore diverges.

Other Tests You Could Consider

While the Integral Test is a great fit here, there are other tests you could potentially use or consider to confirm your solution:

Comparison Tests: You could try comparing this series to other series whose convergence or divergence you already know. However, finding a simple series to compare it to might be challenging.
Limit Comparison Test: This test can be useful, but you'd need to carefully choose a comparison series to make it work effectively.

Key Takeaways

Understanding the difference between convergence and divergence is critical in series analysis.
The Integral Test is super helpful for series involving functions that are easy to integrate.
Always check the conditions (continuous, positive, and decreasing) before applying the Integral Test.
If the integral diverges, the series also diverges.

Further Exploration

Want to dig deeper? Here are some ideas:

Practice with other series! Try to determine if different series converge or diverge.
Explore other convergence tests like the Ratio Test and the Root Test.
Look into the applications of convergent and divergent series in real-world problems.

Keep practicing, guys! The more you work with these concepts, the more comfortable you'll become. Math can be challenging, but it's also incredibly rewarding when you finally understand a concept. Happy calculating! This is a great exercise, and understanding this helps with other problems.

Additional Tips for Series Problems

Let's add some extra tips to help you tackle these types of problems with more confidence. When you are confronted with series convergence or divergence questions, follow these steps to make sure you get the best result:

Identify the Series Type: First and foremost, recognize the type of series. Is it a p-series, a geometric series, or something else? Knowing the type can guide you towards the appropriate tests. For example, the Ratio Test is particularly handy for series that involve factorials or exponential terms. Geometric series have a simple convergence criterion.
Simplify and Rewrite: Sometimes, you might need to rewrite the series to make it more manageable. This could involve simplifying terms, factoring out constants, or breaking down complex expressions. The goal is to get a clearer picture of the series' behavior. A little algebraic manipulation can go a long way in clarifying whether a series will converge or diverge.
Choose the Right Test: Select the most appropriate test based on the series' characteristics. The Integral Test is good for functions that are easy to integrate and whose behavior (increasing/decreasing) is clear. The Ratio Test is excellent for dealing with factorials. The Comparison Tests are useful when you can relate your series to a series whose convergence/divergence is already known. Don't be afraid to experiment with different tests until you find one that works!
Check Conditions: Always verify that the conditions for the chosen test are met. For example, the Integral Test requires the function to be continuous, positive, and decreasing. The Comparison Tests require you to establish the correct inequality. Skipping this step can lead to incorrect conclusions, so double-check those conditions!
Evaluate Carefully: Once you've chosen a test and checked the conditions, execute the test correctly. This might involve evaluating an integral, calculating a limit, or comparing terms. Make sure your calculations are accurate and that you're following the steps of the test precisely. A small calculation error can derail the whole process.
State Your Conclusion: Clearly state whether the series converges or diverges and provide a brief explanation based on your test results. Be confident in your answer! Remember to always mention the test you used. This ensures your solution is understandable and complete.
Practice Regularly: The best way to master series convergence and divergence is through consistent practice. Work through a variety of problems, and don't be discouraged if you struggle at first. The more you practice, the more familiar you will become with different types of series and tests. Regularly working on problems will help you recognize patterns and apply the tests more effectively.

Advanced Tips and Tricks

Let's amp up your knowledge with some advanced strategies and tricks to make you a series whiz!

Recognize Common Series: Familiarize yourself with common series and their convergence properties. This includes geometric series ( $\sum ar^{n-1}$ ), p-series ( $\sum \frac{1}{n^{p}}$ ), and telescoping series. Knowing these can help you quickly identify the behavior of a series. Often, a quick glance can help you eliminate less effective approaches.
Use Limit Properties Strategically: Utilize limit properties to simplify expressions and make them easier to analyze. For example, if you encounter a series with terms that include both polynomials and exponentials, the limit properties can help simplify your work. Remember, exponential functions tend to dominate polynomial functions as n approaches infinity.
Combine Tests: Don't be afraid to combine different tests if necessary. Sometimes, you might use one test to get an initial idea of the series' behavior and then use another test to confirm or refine your conclusion. Flexibility is key! Combining tests can provide a more comprehensive approach.
Handle Alternating Series Carefully: Be extra cautious when dealing with alternating series (those with terms that alternate in sign). Remember to use the Alternating Series Test, which requires checking that the absolute values of the terms decrease monotonically to zero. Alternating series are easy to get wrong because of the extra conditions.
Understand Remainder Estimates: If you're studying Taylor and Maclaurin series, become familiar with remainder estimates (e.g., Lagrange error bound). These allow you to determine the accuracy of your approximations and bound the error. This is a very valuable concept that links series to approximations.
Look for Telescoping Series: Telescoping series are those where most terms cancel out, leaving only a few. Recognizing this pattern can simplify calculations significantly. Look for series where you can rewrite the general term as the difference of two consecutive terms. These series are typically easy to solve once you spot the cancellation pattern.
Master the Ratio and Root Tests: These tests are particularly useful for series involving factorials or exponential functions. The Ratio Test helps when you're working with factorials or products of terms. The Root Test is effective when dealing with nth powers. Getting comfortable with these will make you more versatile.
Practice with Challenging Problems: Seek out more complex problems to challenge your understanding. Look for series that combine different types of terms, require multiple steps, or involve more subtle manipulations. The more problems you solve, the more skills you will master!

With these extra tips and tricks, you will be well-equipped to tackle any series convergence or divergence problem. Keep practicing, stay curious, and enjoy the journey of discovery! Math is a beautiful and rewarding field, and the ability to understand infinite series is a powerful skill. Keep learning, keep exploring, and keep having fun with math! Good luck, and happy calculating! Now go out there and conquer those series! You got this!