Worked example: limit comparison test | Series | AP Calculus BC | Khan Academy

Khan Academy
26 Dec 201604:49
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TLDRIn this instructional video, the presenter explains the Limit Comparison Test for series convergence. The test involves comparing the behavior of two series as n approaches infinity. If the limit of the ratio of their terms is a positive constant, both series converge or diverge together. The example given illustrates this by comparing a geometric series with a similar series, concluding that if the geometric series converges, the original series does as well, based on the test's criteria.

Takeaways
  • 🔍 The limit comparison test is used to determine whether a series converges or diverges.
  • 🔗 Two series, denoted as a sub-n and b sub-n, must have terms that are greater than or equal to zero.
  • 🔢 If the limit of (a sub-n / b sub-n) as n approaches infinity equals a positive constant, then either both series converge or both diverge.
  • 🔄 The test implies that if two series behave similarly for large values of n, they share the same convergence or divergence property.
  • 🔍 The goal is to compare a sub-n with another series, b sub-n, that has similar behavior as n becomes large.
  • 📉 The series a sub-n, 2^n / (3^n - 1), is compared to b sub-n, 2^n / 3^n, to use the limit comparison test.
  • ➗ By simplifying the expression, the limit of (a sub-n / b sub-n) as n approaches infinity is found to be 1.
  • 🟰 The limit of 1 indicates both series converge or diverge together, as 1 is between 0 and infinity.
  • 📉 The series b sub-n is a geometric series with a common ratio less than 1, so it converges.
  • ✅ Since b sub-n converges, the original series a sub-n also converges by the limit comparison test.
Q & A

  • What is the limit comparison test?

    -The limit comparison test is a method used to determine the convergence of a series by comparing it to another series. If the limit of the ratio of the terms of the two series as n approaches infinity is a positive constant, then both series either converge or diverge together.

  • What are the conditions for using the limit comparison test?

    -The limit comparison test can be used when both series have terms greater than or equal to zero for all n. Additionally, the limit of the ratio of the terms of the two series as n approaches infinity must be a positive constant.

  • Why is the behavior of terms as n approaches infinity important in the limit comparison test?

    -The behavior of terms as n approaches infinity is crucial because it indicates whether the series will converge or diverge. If the terms of the two series behave similarly as n gets large, it suggests that they will converge or diverge together.

  • What does it mean if the limit of the ratio of terms is greater than zero?

    -If the limit of the ratio of terms is greater than zero, it means that the series are tied in their convergence or divergence. Both series will either converge or diverge together.

  • What series was suggested to compare with the given series in the script?

    -The series suggested for comparison in the script is a geometric series with terms of the form \(2^n / 3^n\).

  • How does the comparison series relate to the original series in the script?

    -The comparison series is similar to the original series in the script, with the only difference being the presence of a minus one in the denominator of the original series. As n gets large, this difference becomes negligible, making the series behave similarly.

  • What is the algebraic manipulation performed in the script to find the limit?

    -The algebraic manipulation involves dividing the numerator and the denominator by \(2^n\) and then by \(3^n\), which simplifies the expression to \(1 / (1 - 1/3^n)\). As n approaches infinity, \(1/3^n\) approaches zero, making the limit equal to one.

  • What is the significance of the limit approaching one in the context of the limit comparison test?

    -A limit approaching one indicates that the ratio of the terms of the two series is constant and positive, which means the series are tied in their convergence or divergence. Since the limit is between zero and infinity, both series converge or diverge together.

  • Why is the geometric series chosen for comparison considered to converge?

    -The geometric series is considered to converge because its common ratio is less than one, which is a condition for convergence in geometric series.

  • What conclusion is drawn about the original series S based on the limit comparison test?

    -Based on the limit comparison test, since the comparison series converges, the original series S also converges.

Outlines
00:00
📚 Introduction to Limit Comparison Test

The paragraph introduces the limit comparison test, a method used to determine the convergence of two series. It explains that if two series have terms a_n and b_n, both non-negative, and the limit of a_n/b_n as n approaches infinity is a positive constant, then both series converge or diverge together. The paragraph emphasizes the importance of the behavior of the series as n becomes very large, suggesting that if the terms behave similarly, their convergence properties will also be similar.

Mindmap
Keywords
💡Limit Comparison Test
The Limit Comparison Test is a method used in calculus to determine the convergence of an infinite series by comparing it to another series whose convergence is already known. In the video, the instructor explains that if the limit of the ratio of the terms of two series is a positive constant, then both series converge or diverge together. This is central to the video's theme of testing series convergence.
💡Convergence
Convergence in the context of series refers to the property where the sum of an infinite series approaches a finite value as the number of terms increases. The video discusses the convergence of a series as a key outcome of applying the Limit Comparison Test, using it to conclude that the series under discussion converges because a comparable series, known to be convergent, was identified.
💡Divergence
Divergence is the opposite of convergence; it describes a situation where the sum of an infinite series does not approach a finite value but instead grows without bound. The script mentions that if the limit comparison test indicates a positive constant, both series either converge or diverge, highlighting the importance of divergence in understanding series behavior.
💡Series
A series in mathematics is the sum of the terms of a sequence. The script discusses different series, denoted by 'a sub-n' and 'b sub-n', to illustrate the application of the Limit Comparison Test. The series are integral to the video's narrative as they are the subjects being tested for convergence or divergence.
💡Positive Constant
A positive constant is a value that is greater than zero but less than infinity. In the script, the instructor refers to the limit of the ratio of terms from two series being a positive constant as a condition for the Limit Comparison Test to conclude that both series converge or diverge together.
💡Algebraic Manipulation
Algebraic manipulation involves the use of algebraic techniques to simplify or transform expressions. The instructor in the video uses algebraic manipulation to simplify the ratio of terms from the series to find the limit, which is a crucial step in applying the Limit Comparison Test.
💡Geometric Series
A geometric series is a series where each term after the first is found by multiplying the previous term by a constant called the common ratio. The script identifies a series as geometric and uses the fact that its common ratio is less than one to conclude that it converges, which is a key part of the argument for the original series' convergence.
💡Common Ratio
The common ratio in a geometric series is the factor by which we multiply each term to get the next. The video mentions that the common ratio of the identified geometric series is less than one, which is a condition that ensures the series converges, a critical point in the video's conclusion about the original series.
💡Numerator and Denominator
In a fraction, the numerator is the top number and the denominator is the bottom number. The script discusses dividing both the numerator and the denominator by the same term to simplify the expression when finding the limit, a technique used to apply the Limit Comparison Test effectively.
💡Limit
In calculus, a limit is the value that a function or sequence approaches as the input approaches some value. The video script repeatedly refers to finding the limit as 'n' approaches infinity to determine the behavior of the series terms, which is essential for the Limit Comparison Test.
💡Infinity
Infinity is a concept that refers to an unbounded quantity that is larger than any number. The script uses the term 'infinity' to describe the behavior of the series terms as 'n' becomes extremely large, which is a fundamental aspect of analyzing series convergence or divergence.
Highlights

Introduction to the Limit Comparison Test for determining convergence of series.

Explanation of the conditions for the Limit Comparison Test: both series must be greater than or equal to zero for all n.

The importance of the limit as n approaches infinity of a_n/b_n being a positive constant for the test to apply.

Behavior of series as n gets large and how it influences their convergence or divergence.

Identifying a suitable series for comparison that behaves similarly as n approaches infinity.

Algebraic manipulation to simplify the expression for the limit comparison.

Cancellation of terms in the numerator and denominator to simplify the limit expression.

Understanding the behavior of the limit as n approaches infinity and its approach to one.

The conclusion that if the limit is a positive constant between zero and infinity, both series converge or diverge together.

Application of the Limit Comparison Test to a specific series and its comparison to a geometric series.

Identification of the common ratio in the geometric series and its significance in determining convergence.

Final conclusion that the original series S converges based on the Limit Comparison Test.

The practical application of the Limit Comparison Test in analyzing the convergence of series.

The importance of choosing the right series for comparison in the Limit Comparison Test.

The role of algebraic manipulation in making the comparison process more straightforward.

The significance of the limit approaching one in determining the convergence of the series.

The instructional approach to explaining the Limit Comparison Test and its application.

Transcripts

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Series AnalysisLimit TestConvergenceMathematicsEducationalAlgebraic ManipulationGeometric SeriesInfinite LimitsCalculus ConceptsLearning Resource