Calc 2 Comparison Tests
37 flashcards covering Calc 2 Comparison Tests for the CALCULUS-2 Calc 2 Topics section.
The Comparison Tests in Calculus II are methods used to determine the convergence or divergence of infinite series. These tests are defined in standard calculus curricula and are essential for students to master as they build on concepts from earlier courses. The two primary tests are the Direct Comparison Test and the Limit Comparison Test, which allow students to compare a given series to a known benchmark series to draw conclusions about its behavior.
In practice exams and competency assessments, questions related to Comparison Tests often present a series and ask students to identify whether it converges or diverges by comparing it to a simpler series. A common pitfall is misapplying the tests, particularly in cases where the chosen comparison series does not satisfy the necessary conditions for the test to be valid. Students may also overlook the importance of ensuring that both series are positive for the Direct Comparison Test to be applicable. Remember to carefully check the conditions before proceeding with your comparisons.
Terms (37)
- 01
What is the purpose of the Comparison Test in series convergence?
The Comparison Test is used to determine the convergence or divergence of a series by comparing it to a known convergent or divergent series (Stewart Calculus, series chapter).
- 02
When can the Limit Comparison Test be applied?
The Limit Comparison Test can be applied when both series have positive terms and the limit of the ratio of their terms approaches a positive finite number (Stewart Calculus, series chapter).
- 03
What is the outcome if the series an converges and bn is a larger series?
If an converges and bn is larger, then bn must also converge, confirming that both series converge (Stewart Calculus, series chapter).
- 04
Under what condition does the Comparison Test fail?
The Comparison Test fails if the series being compared does not converge or diverge, meaning it cannot be determined from the comparison (Stewart Calculus, series chapter).
- 05
What is the first step in using the Comparison Test?
The first step is to identify a known convergent or divergent series to compare with the series in question (Stewart Calculus, series chapter).
- 06
How do you determine the convergence of the series 1/n^2 using the Comparison Test?
You can compare it to the convergent p-series 1/n^p with p=2, showing that it converges by the Comparison Test (Stewart Calculus, series chapter).
- 07
What happens if the series an diverges and bn is a smaller series?
If an diverges and bn is smaller, you cannot conclude anything about bn's convergence; it may converge or diverge (Stewart Calculus, series chapter).
- 08
What series is often used as a comparison for the harmonic series?
The series 1/n is often used to compare with the harmonic series, which diverges (Stewart Calculus, series chapter).
- 09
When using the Limit Comparison Test, what should the limit of an/bn equal?
The limit of an/bn should equal a positive finite number for the test to be valid (Stewart Calculus, series chapter).
- 10
What type of series does the Comparison Test apply to?
The Comparison Test applies to series with positive terms, where the terms are non-negative (Stewart Calculus, series chapter).
- 11
What is the conclusion if the Limit Comparison Test results in zero?
If the limit approaches zero, the series an converges if bn converges, but no conclusion can be drawn if bn diverges (Stewart Calculus, series chapter).
- 12
What is the relationship between the terms of two series in the Limit Comparison Test?
The terms of both series must be positive, and the limit of their ratio must exist and be a positive finite number (Stewart Calculus, series chapter).
- 13
How does the Direct Comparison Test differ from the Limit Comparison Test?
The Direct Comparison Test compares series directly, while the Limit Comparison Test uses the limit of the ratio of their terms (Stewart Calculus, series chapter).
- 14
What is a common series used to demonstrate divergence in the Comparison Test?
The series 1/n is commonly used to demonstrate divergence when compared with other series (Stewart Calculus, series chapter).
- 15
What is the significance of the p-series in the Comparison Test?
P-series, defined as 1/n^p, are significant as they provide a benchmark for convergence based on the value of p (Stewart Calculus, series chapter).
- 16
What is the conclusion if both series in the Limit Comparison Test diverge?
If both series diverge, then the conclusion is that the series an also diverges (Stewart Calculus, series chapter).
- 17
What condition must be met for the Direct Comparison Test to be valid?
For the Direct Comparison Test to be valid, the series being compared must have terms that are non-negative (Stewart Calculus, series chapter).
- 18
Which series converges: 1/n^3 or 1/n^2?
The series 1/n^3 converges, while 1/n^2 diverges, demonstrating how p-series behave differently based on p's value (Stewart Calculus, series chapter).
- 19
What should you check first when applying the Comparison Test?
You should first check if the series has positive terms before applying the Comparison Test (Stewart Calculus, series chapter).
- 20
How can you use the Comparison Test to show that 1/(n^2 + n) converges?
You can compare it with 1/n^2, which is a convergent p-series, to conclude that 1/(n^2 + n) also converges (Stewart Calculus, series chapter).
- 21
What is the relationship between the convergence of an and bn in the Limit Comparison Test?
If the limit of an/bn is a positive finite number, then both series either converge or diverge together (Stewart Calculus, series chapter).
- 22
What is the role of the constant factor in the Comparison Test?
A constant factor does not affect the convergence or divergence of the series when using the Comparison Test (Stewart Calculus, series chapter).
- 23
What is the conclusion if an is less than bn and bn converges?
If an is less than bn and bn converges, then an also converges by the Comparison Test (Stewart Calculus, series chapter).
- 24
What type of series is 1/n^p with p > 1?
The series 1/n^p with p > 1 is a convergent p-series (Stewart Calculus, series chapter).
- 25
How do you determine if 1/(n^2 + 1) converges?
You can compare it to 1/n^2, which converges, thus showing that 1/(n^2 + 1) also converges (Stewart Calculus, series chapter).
- 26
What is an example of a divergent series?
An example of a divergent series is the harmonic series, 1/n, which diverges (Stewart Calculus, series chapter).
- 27
What is the conclusion if an diverges and bn is larger?
If an diverges and bn is larger, then bn must also diverge (Stewart Calculus, series chapter).
- 28
What does it mean if the limit of an/bn equals infinity?
If the limit equals infinity, bn must diverge if an diverges, confirming divergence (Stewart Calculus, series chapter).
- 29
What condition is necessary for the Direct Comparison Test?
The necessary condition for the Direct Comparison Test is that both series must have non-negative terms (Stewart Calculus, series chapter).
- 30
What is the significance of the ratio test in relation to the Comparison Test?
The ratio test provides another method to determine convergence, but it is not directly related to the Comparison Test (Stewart Calculus, series chapter).
- 31
How can the Comparison Test be applied to alternating series?
The Comparison Test can be applied to alternating series by comparing the absolute values of their terms (Stewart Calculus, series chapter).
- 32
What happens if a series diverges but is compared to a convergent series?
If a series diverges but is compared to a convergent series, no conclusion can be drawn about the divergence of the original series (Stewart Calculus, series chapter).
- 33
What is the conclusion if both series converge in the Limit Comparison Test?
If both series converge, then the series an converges as well (Stewart Calculus, series chapter).
- 34
What is the outcome if the comparison series diverges?
If the comparison series diverges, the original series may also diverge, but further investigation is needed (Stewart Calculus, series chapter).
- 35
What is the role of the positive constant in series comparison?
A positive constant can be factored out and does not affect the convergence or divergence of the series (Stewart Calculus, series chapter).
- 36
What is the conclusion if an is greater than bn and bn converges?
If an is greater than bn and bn converges, no conclusion can be drawn about an's convergence (Stewart Calculus, series chapter).
- 37
What is a necessary condition for applying the Limit Comparison Test?
A necessary condition is that both series must have positive terms for the Limit Comparison Test to be applicable (Stewart Calculus, series chapter).