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Comparison & limit comparison test

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  1. Overview:

  2. Comparison test
  3. Limit Comparison test
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  1. Convergence & Divergence of Comparison Tests
    Use the Comparison Test to determine if the series converge or diverge.
    1. ∑n=1∞12n+5 \sum_{n=1}^{\infty}\frac{1}{2^n+5}
    2. ∑n=1∞n4+5n5−sin4(2n) \sum_{n=1}^{\infty}\frac{n^4+5}{n^5-sin^4(2n)}
    3. ∑n=1∞n4cos4(7n)−1n6 \sum_{n=1}^{\infty}\frac{n^4cos^4(7n)-1}{n^6}
  2. Convergence & Divergence of Limit Comparison Tests
    Use the Limit Comparison Test to determine if the series converge or diverge.
    1. ∑n=3∞n2+n3n8+n4 \sum_{n=3}^{\infty}\frac{n^2+n^3}{\sqrt{n^8+n^4}}
    2. ∑n=1∞1n2−7n−12 \sum_{n=1}^{\infty}\frac{1}{n^2-7n-12}
  3. Understanding of Both Tests
    Use both the comparison and limit comparison test for the series ∑k=1∞k3−1k3−2k2+5\sum_{k=1}^{\infty}\frac{\sqrt{k^3-1}}{k^3-2k^2+5} . What do both tests say?
    Topic Notes
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    You may notice that some series look very complicated, but it shares the same properties as another series that looks very simple and easy. In this case, we can use the comparison test or limit comparison test. We will look at what conditions must be met to use these tests, and then use the tests on some complicated looking series. Lastly, we will use both the comparison test and the limit comparison test on a series, and conclude that they give the same result.

    Introduction to Comparison and Limit Comparison Tests

    The comparison test and limit comparison test are powerful tools in calculus for evaluating the divergence of complex series. These tests work by comparing a given series to simpler, well-understood series. The introduction video provides a crucial foundation for grasping these concepts. In the comparison test, we directly compare terms of two series, while the limit comparison test examines the limit of the ratio of corresponding terms. Both methods are particularly useful when dealing with series that are challenging to evaluate using other techniques. By comparing an unknown series to a known one, we can often determine its behavior more easily. These tests are especially valuable for positive term series and can help in situations where direct evaluation is difficult. Understanding these tests enhances one's ability to analyze and solve problems involving infinite series, making them essential tools in advanced calculus and mathematical analysis.

    When divergence of complex series is considered, the comparison test can be particularly insightful. It allows us to draw conclusions about the behavior of a series by leveraging our understanding of another series. This method is not only efficient but also simplifies the process of analyzing infinite series that might otherwise be too complex to handle directly.

    Understanding the Comparison Test

    The comparison test is a powerful tool in the realm of calculus, specifically used for determining the convergence or divergence of infinite series. This test relies on comparing the given series to another series whose behavior is already known. By establishing a relationship between the two series, we can draw conclusions about the convergence or divergence of the series in question.

    To apply the comparison test, we need two series: the series we want to analyze and a reference series with known convergence properties. Let's call our series of interest an and the reference series bn. The key is to compare the terms of these series for all values of n greater than or equal to some starting point N.

    There are two main scenarios in the comparison test:

    1. If 0 an bn for all n N, and bn is convergent, then an is also convergent.

    2. If an bn 0 for all n N, and bn is divergent, then an is also divergent.

    The logic behind this test is intuitive. In the first case, if the larger series converges, the smaller series must also converge. In the second case, if the smaller series diverges, the larger series must also diverge.

    To apply the comparison test effectively, follow these steps:

    1. Identify a suitable reference series bn with known convergence behavior.

    2. Establish an inequality between an and bn for all n N.

    3. Verify that the inequality holds for all terms beyond the Nth term.

    4. Draw a conclusion based on the convergence or divergence of the reference series.

    Let's consider an example to illustrate this process. Suppose we want to determine the convergence of the series (1/n²). We can compare this to the p-series (1/n) which we know diverges for p 1 and converges for p > 1. In this case, 1/n² 1/n for all n 1, and we know that (1/n) diverges. However, we can't conclude anything about our series from this comparison.

    Instead, let's compare it to (1/n1.5), which we know converges. We can see that 1/n² 1/n1.5 for all n 1. Since the larger series (1/n1.5) converges, we can conclude that our series (1/n²) also converges.

    When using the comparison test, it's crucial to avoid common pitfalls:

    1. Ensure that the inequality holds for all terms beyond a certain point, not just for a few initial terms.

    2. Be careful not to reverse the inequality. Remember, for convergence, we need the series in question to be smaller than or equal to a convergent series.

    3. Don't compare to a series with unknown convergence behavior. The reference series must have a known convergence or divergence.

    4. Avoid comparing series with alternating signs using this test. The comparison test works best with series of positive terms.

    The comparison test is particularly useful when dealing with series that resemble known convergent or divergent series. For instance, when analyzing series involving factorials, exponentials, or powers, you can often find a suitable p-series

    Applying the Comparison Test

    The comparison test is a powerful tool for evaluating the convergence or divergence of series. This step-by-step guide will help you apply the comparison test to different types of series, with examples of both convergent and divergent comparisons.

    Step 1: Understand the Comparison Test

    The comparison test states that if 0 an bn for all n N, where N is some positive integer, then:

    • If Σbn converges, then Σan also converges.
    • If Σan diverges, then Σbn also diverges.

    Step 2: Choose an Appropriate Comparison Series

    Select a series with known convergence behavior that is similar to your given series. Common comparison series include:

    • p-series: Σ(1/np), which converges for p > 1 and diverges for p 1
    • Geometric series: Σarn, which converges for |r| < 1 and diverges for |r| 1
    • Harmonic series: Σ(1/n), which diverges

    Step 3: Establish the Inequality

    Show that the terms of your series are less than or equal to (for convergence) or greater than or equal to (for divergence) the terms of the comparison series for all n N.

    Step 4: Apply the Comparison Test

    Use the known behavior of the comparison series to draw conclusions about your given series.

    Example 1: Convergent Comparison

    Consider the series Σ(1/(n2 + 1)). We can compare this to the p-series Σ(1/n2).

    1/(n2 + 1) 1/n2 for all n 1

    Since Σ(1/n2) converges (p-series with p = 2 > 1), we conclude that Σ(1/(n2 + 1)) also converges.

    Example 2: Divergent Comparison

    Consider the series Σ(1/(n + 1)). We can compare this to the harmonic series Σ(1/n).

    1/(n + 1) 1/(n + n) = 1/(2n) for all n 1

    Since Σ(1/(2n)) = (1/2)Σ(1/n) diverges (constant multiple of harmonic series), we conclude that Σ(1/(n + 1)) also diverges.

    Step 5: Interpret the Results

    Based on the comparison, you can determine whether your series converges or diverges. Remember that the comparison test only provides conclusive results in certain cases:

    • If the comparison series converges and your series terms are smaller, your series converges.
    • If the comparison series diverges and your series terms are larger, your series diverges.
    • If the inequality goes the other way, the test is inconclusive, and you may need to try a different comparison or use another convergence test.

    Introduction to the Limit Comparison Test

    The limit comparison test is a powerful tool in calculus and mathematical analysis, specifically designed to determine the convergence or divergence of infinite series. This test is particularly useful when dealing with series that are challenging to evaluate using simpler methods, such as the ratio test or the root test. The limit comparison test shines in situations where we have a complex series that behaves similarly to a simpler, well-understood series.

    To apply the limit comparison test, certain conditions must be met. First, we need two series: the series we want to analyze and a comparison series whose behavior is known. Both series must have positive terms for all sufficiently large n. This requirement ensures that we're dealing with series that are amenable to comparison.

    The core concept of the limit comparison test revolves around examining the limit of the ratio of corresponding terms from the two series as n approaches infinity. This ratio provides insight into how the series behave relative to each other in the long run. Mathematically, we express this as:

    lim(n) (a_n / b_n) = L

    Where a_n represents the general term of the series we're investigating, and b_n is the general term of the comparison series. The value of L, the limit of this ratio, determines the outcome of the test.

    There are three possible scenarios when applying the limit comparison test:

    1. If 0 < L < (L is finite and positive), both series either converge or diverge together. This means that if we know the behavior of the comparison series, we can immediately conclude the same for our original series.

    2. If L = 0, and the comparison series converges, then our original series also converges. This scenario indicates that the terms of our series become significantly smaller than those of the converging comparison series, ensuring convergence.

    3. If L = , and the comparison series diverges, then our original series also diverges. In this case, the terms of our series grow much faster than those of the diverging comparison series, leading to divergence.

    It's important to note that if L = 0 and the comparison series diverges, or if L = and the comparison series converges, the test is inconclusive. In these cases, we need to employ other methods to determine the series' behavior.

    The limit comparison test is particularly effective when dealing with series that involve complex expressions, such as those with exponentials, logarithms, or trigonometric functions. By comparing these to simpler series like p-series or geometric series, we can often draw conclusions about their convergence or divergence without having to directly sum or manipulate the original series.

    When applying the limit comparison test, choosing an appropriate comparison series is crucial. The ideal comparison series should have a similar structure to the original series but be simpler to analyze. Common choices include p-series (1/n^p), geometric series (ar^n), or exponential series (r^n/n!), depending on the nature of the original series.

    In practice, the limit comparison test often simplifies the analysis of series by allowing us to focus on the dominant terms as n approaches infinity. This approach can reveal the essential behavior of a series without getting bogged down in the complexities of its exact form.

    To successfully apply the limit comparison test, it's essential to have a strong grasp of limit calculations and a good intuition for series behavior. Familiarity with common series types and their convergence properties is also beneficial, as it helps in selecting appropriate comparison series.

    In conclusion, the limit comparison test is a versatile and powerful tool in the study of infinite series. By comparing the asymptotic behavior of two series, it provides a method to determine convergence or divergence in cases where other tests may fail or be more difficult to apply. Its ability to handle complex series by relating them to simpler, well-understood series makes it an indispensable technique in advanced calculus and analysis.

    Applying the Limit Comparison Test

    The limit comparison test is a powerful tool in calculus for determining the convergence or divergence of infinite series. This test is particularly useful when dealing with complex series that are difficult to evaluate using other methods. In this section, we'll explore how to apply the limit comparison test, interpret its results, and use it effectively in various scenarios.

    To begin, let's outline the steps for applying the limit comparison test:

    1. Choose a series with known convergence behavior to compare with the given series.
    2. Form the ratio of the general terms of the two series.
    3. Calculate the limit of this ratio as n approaches infinity.
    4. Interpret the result to determine convergence or divergence.

    Let's consider an example to illustrate these steps. Suppose we want to determine the convergence of the series Σ(n^2 / (n^3 + 1)). We can compare this to the p-series Σ(1/n), which we know converges for p > 1.

    Step 1: We choose Σ(1/n) as our comparison series.
    Step 2: We form the ratio (n^2 / (n^3 + 1)) / (1/n).
    Step 3: We simplify and calculate the limit: lim(n) (n^3 / (n^3 + 1)) = lim(n) (1 / (1 + 1/n^3)) = 1

    Step 4: Interpreting the result, we find that the limit is a positive finite number (1). This means our original series behaves similarly to Σ(1/n), which diverges. Therefore, our series also diverges.

    Now, let's discuss how to interpret different results of the limit comparison test:

    • If the limit is zero: The original series converges if the comparison series converges.
    • If the limit is a positive finite number: The original series converges if and only if the comparison series converges.
    • If the limit is infinity: The original series diverges if the comparison series diverges.

    It's important to note that if the limit doesn't exist or is negative, the test is inconclusive, and other methods should be used.

    The limit comparison test is particularly useful for complex series that involve algebraic or transcendental functions. For instance, consider the series Σ(sin(n) / n^2). We can compare this to Σ(1/n^2), which we know converges. The limit of their ratio as n approaches infinity is 0, indicating that our original series converges.

    When dealing with series involving exponential functions, such as Σ(e^(-n) / n), we might compare it to a geometric series like Σ(1/2^n). The limit comparison test can help us determine convergence in cases where other tests might be more challenging to apply.

    It's worth noting that the choice of comparison series is crucial. A well-chosen comparison can simplify the analysis significantly. Generally, we look for a simpler series that behaves similarly to our given series as n approaches infinity.

    The limit comparison test can also be used to determine divergence. For example, if we're examining Σ(n / (n^2 + 1)), we might compare it to Σ(1/n). The limit of their ratio is 1, and since Σ(1/n) diverges, our original series must also diverge.

    In conclusion, the limit comparison test is a versatile and powerful tool for evaluating the convergence or divergence of infinite series. By comparing the behavior of a given series to a known series, we can often simplify our analysis and draw conclusions about convergence. This test is particularly valuable when dealing with series that involve complicated functions or when other convergence tests prove inconclusive. Mastering the application of the limit comparison test enhances our ability to analyze a

    Comparing the Comparison and Limit Comparison Tests

    When analyzing the convergence or divergence of infinite series, mathematicians often employ various tests to determine the behavior of these series. Two commonly used techniques are the comparison test and the limit comparison test. While both tests serve similar purposes, they have distinct characteristics and applications that make them suitable for different scenarios.

    The comparison test, also known as the direct comparison test, involves comparing the terms of a given series to those of a known convergent or divergent series. This test is particularly useful when dealing with series that have similar structures or behaviors. On the other hand, the limit comparison test focuses on the ratio of corresponding terms from two series as the index approaches infinity.

    One key advantage of the comparison test is its simplicity and directness. When a series can be easily compared to a well-known convergent or divergent series, this test provides a straightforward approach to determining convergence. For instance, when dealing with series involving simple functions like polynomials or exponentials, the comparison test often proves to be the most efficient method.

    However, the comparison test may fall short when dealing with more complex series or when finding an appropriate comparison series becomes challenging. In such cases, the limit comparison test often comes to the rescue. This test is particularly powerful when dealing with series that have similar asymptotic behavior but may not be directly comparable term by term.

    The limit comparison test shines in scenarios where the ratio of corresponding terms from two series approaches a finite, non-zero limit. This property makes it especially useful for series involving rational functions, logarithms, or more intricate combinations of functions. Additionally, the limit comparison test can sometimes provide insights into the rate of convergence or divergence, which the direct comparison test may not reveal.

    To illustrate the application of both tests, let's consider the series Σ(1/n^2) from n=1 to infinity. Using the comparison test, we can compare this series to the well-known p-series Σ(1/n^p) with p=2, which is known to converge. Since the terms of our series are identical to the p-series, we can conclude that Σ(1/n^2) converges.

    Applying the limit comparison test to the same series, we can compare it to Σ(1/n) from n=1 to infinity (the harmonic series). Taking the limit of the ratio of corresponding terms as n approaches infinity: lim(n) [(1/n^2) / (1/n)] = lim(n) (1/n) = 0. Since this limit exists and is finite, we conclude that both series behave similarly. As the harmonic series diverges, our original series Σ(1/n^2) must converge.

    In this example, both tests lead to the same conclusion, demonstrating their consistency. However, the comparison test provided a more direct and intuitive approach, while the limit comparison test offered a different perspective on the series' behavior.

    Another scenario where both tests can be applied is the series Σ(n/(n^2+1)) from n=1 to infinity. Using the comparison test, we can compare this series to Σ(1/n) from n=1 to infinity. Since n/(n^2+1) < 1/n for all n1, and the harmonic series diverges, we conclude that our series also diverges.

    Applying the limit comparison test to the same series, we compare it again to Σ(1/n). Taking the limit of the ratio: lim(n) [(n/(n^2+1)) / (1/n)] = lim(n) [n^2/(n^2+1)] = 1. As this limit exists and is non-zero, both series behave similarly, and since the harmonic series diverges, our original series must also diverge.

    In conclusion, while the comparison test and limit comparison test often lead to the same results, their applicability and ease of use can vary depending on the series in question. The comparison test excels in simplicity and directness for series with clear comparisons, while the limit comparison test offers a powerful tool for analyzing more complex series with similar asymptotic behaviors. Mastering both techniques equ

    Conclusion and Further Practice

    The comparison and limit comparison tests are powerful tools in mathematical analysis for determining series convergence. These tests allow us to evaluate complex series by comparing them to simpler, known series. The introduction video provides a crucial foundation for understanding these concepts, demonstrating their application and limitations. To truly master these tests, it's essential to practice applying them to a variety of series. This hands-on experience will help solidify your understanding and improve your problem-solving skills. We encourage you to seek out additional resources and practice problems to further your study of series convergence tests. Online platforms, textbooks, and academic websites offer a wealth of exercises ranging from basic to advanced levels. Remember, proficiency in these tests not only enhances your mathematical abilities but also prepares you for more advanced topics in calculus and analysis. Keep practicing, and don't hesitate to revisit the introductory material as needed to reinforce your understanding.

    Overview:

    Comparison test

    Step 1: Introduction to the Comparison Test

    The comparison test is a method used in mathematical analysis to determine the convergence or divergence of an infinite series. The basic idea is to compare a given series with another series whose convergence properties are already known. This method is particularly useful when it is difficult to determine the convergence of the original series directly.

    Step 2: Definition and Conditions

    The comparison test involves two series, denoted as ana_n and bnb_n. The conditions for applying the comparison test are as follows:

    • Both series ana_n and bnb_n must be non-negative for all nn, i.e., an≥0a_n \geq 0 and bn≥0b_n \geq 0.
    • There must exist a relationship between the terms of the two series such that an≤bna_n \leq b_n for all nn.

    Step 3: Applying the Comparison Test

    Once the conditions are met, the comparison test can be applied as follows:

    • If the series bnb_n is convergent, then the series ana_n is also convergent.
    • If the series ana_n is divergent, then the series bnb_n is also divergent.
    This means that if you know the convergence behavior of the larger series bnb_n, you can infer the behavior of the smaller series ana_n.

    Step 4: Practical Example

    To illustrate the comparison test, consider the following example:

    • Suppose you have a series ana_n that is difficult to analyze directly.
    • You compare it with another series bnb_n that is known to be convergent.
    • If an≤bna_n \leq b_n for all nn and bnb_n is convergent, then ana_n is also convergent.
    Conversely, if you know that ana_n is divergent and an≤bna_n \leq b_n, then bnb_n must also be divergent.

    Step 5: Important Considerations

    It is crucial to remember the following points when using the comparison test:

    • If the smaller series ana_n is convergent, it does not necessarily mean that the larger series bnb_n is convergent.
    • If the larger series bnb_n is divergent, it does not necessarily mean that the smaller series ana_n is divergent.
    The comparison test only works in one direction: from the known behavior of the larger series to the unknown behavior of the smaller series, and from the known behavior of the smaller series to the unknown behavior of the larger series.

    Step 6: Common Mistakes

    Many students often confuse the conditions and implications of the comparison test. Here are some common mistakes to avoid:

    • Assuming that if the smaller series is convergent, the larger series must also be convergent. This is not true.
    • Assuming that if the larger series is divergent, the smaller series must also be divergent. This is also not true.
    Always ensure that you are comparing the series in the correct direction and that the conditions for the comparison test are strictly met.

    Step 7: Conclusion

    The comparison test is a powerful tool for determining the convergence or divergence of series by comparing them with other series whose behavior is already known. By carefully applying the conditions and understanding the implications, you can effectively use this test to analyze complex series.

    FAQs

    Here are some frequently asked questions about the comparison and limit comparison tests:

    1. What are the conditions for the comparison test?

    The comparison test requires two series with positive terms: the series you want to analyze and a reference series with known convergence. For all n N (where N is some positive integer), you must establish an inequality between the terms of the two series.

    2. What are the conditions for the limit comparison test?

    The limit comparison test requires two series with positive terms for all sufficiently large n. You then examine the limit of the ratio of corresponding terms as n approaches infinity. The test is most useful when this limit exists and is finite and positive.

    3. What does the limit comparison test tell us?

    If the limit L of the ratio of terms is finite and positive (0 < L < ), both series either converge or diverge together. If L = 0 and the comparison series converges, the original series converges. If L = and the comparison series diverges, the original series diverges.

    4. Why is the limit comparison test inconclusive if L = 0 or L = in some cases?

    When L = 0 and the comparison series diverges, or when L = and the comparison series converges, the test doesn't provide enough information to determine the behavior of the original series. In these cases, other methods must be used to analyze the series.

    5. How do you choose an appropriate comparison series?

    Choose a series with known convergence behavior that is similar to your given series. Common choices include p-series, geometric series, or exponential series. The comparison series should have a similar structure but be simpler to analyze than the original series.

    Prerequisite Topics

    Understanding the Comparison & limit comparison test is crucial in calculus, but to truly grasp this concept, it's essential to have a solid foundation in several prerequisite topics. These fundamental concepts provide the necessary context and tools to effectively apply and interpret the comparison and limit comparison tests.

    One of the most important prerequisites is convergence and divergence of infinite series. This topic lays the groundwork for understanding how series behave as they extend infinitely, which is central to the comparison tests. Closely related to this is the geometric series convergence, which provides a specific example of how series can converge or diverge based on their structure.

    Another critical concept is the harmonic series divergence. This serves as a classic example of a divergent series and is often used as a benchmark in comparison tests. Understanding why the harmonic series diverges helps in recognizing the behavior of similar series.

    The ratio test for series and the root test for series are also important prerequisites. These tests provide alternative methods for determining convergence and divergence, and understanding them enhances your ability to choose the most appropriate test for a given series.

    Knowledge of exponential functions in series and logarithmic functions in series is crucial as these types of functions frequently appear in series problems. Their unique properties often play a role in determining convergence or divergence.

    Lastly, familiarity with power series analysis provides insight into how functions can be represented as infinite series. This connection between functions and series is fundamental to many advanced calculus concepts, including the comparison and limit comparison tests.

    By mastering these prerequisite topics, you'll be well-equipped to tackle the intricacies of the Comparison & limit comparison test. Each concept builds upon the others, creating a comprehensive understanding of series behavior. This knowledge not only aids in applying the tests correctly but also in interpreting their results and understanding their limitations. Remember, in mathematics, a strong foundation is key to advanced problem-solving, and these prerequisites form that crucial foundation for mastering series convergence tests.

    Note *The Comparison test says the following:
    Let ∑an\sum a_n and ∑bn\sum b_n be two series where an≤bna_n\leq b_n for all nn and anbn≥0a_nb_n\geq0. Then we say that
    1. If ∑bn\sum b_n is convergent, then ∑an\sum a_n is also convergent
    2. If ∑an\sum a_n is divergent, then ∑bn\sum b_n is also divergent.

    The Limit Comparison Test says the following:
    Let ∑an\sum a_n and ∑bn\sum b_n be two series where an≥0a_n\geq 0 and bnb_n > 0 for all nn. Then we say that

    lim⁡\limn →∞\infty anbn=c\frac{a_n}{b_n}=c

    If cc is a positive finite number, then either both series converge or diverge.
    Basic Concepts
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