Convergence and Divergence - Calculus 2

In summary, the student is preparing for their final exam in Calculus 2 and is struggling with two sections of problems on convergence and divergence/alternating series. They have specific questions for each problem and are seeking confirmation or help with their assumptions. They are familiar with the tests for absolute and conditionally convergent and have a general understanding of most problems. They are also using the comparison test and the divergence test to determine convergence.
  • #1
balloonhf
3
0
Alright, I have my final for Calc 2 on Monday. I am only stuck on two sections of problems because I am terrible at convergence and divergence/alternating series.

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I have questions for each one really. They are as follows:
4)
a. Can I simply factor out the alternating series and apply a test since the alternating portion is constant?
b. Similar to my question from a.
c. Not sure of what test to apply for convergence tests. I have tried both the nth root test and the ratio test and neither work.
d. Similar to my question from c.

5)
a) Again can I factor out the alternating series and say that it is similar to 1/(n)^(1/2) and 1/2<1 so it diverges.
b) and c) - I have literally no idea.
d)Same as a in that I want to know if i can factor out the constant alternating section
e) and f) - No idea

I have a general idea on most problems. Most of my questions are on the same thing that I need confirmed. Also I know the tests for Absolute and Conditionally convergent.

Any sort of help or confirmation on my assumptions is greatly appreciated.
 
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  • #2
balloonhf said:
a. Can I simply factor out the alternating series and apply a test since the alternating portion is constant?
b. Similar to my question from a.

See this .
c. Not sure of what test to apply for convergence tests. I have tried both the nth root test and the ratio test and neither work.

Here you can use the comparison test .
d. Similar to my question from c.

Try the divergence test .

5)
a) Again can I factor out the alternating series and say that it is similar to 1/(n)^(1/2) and 1/2<1 so it diverges.
b) and c) - I have literally no idea.
d)Same as a in that I want to know if i can factor out the constant alternating section

If the non-alternating part is positive and decreasing , then it is sufficient to test whether it goes to 0 as n goes to infinity.

e) and f) - No idea

Try the divergence test on e , and f can be simplified a little ,see how ?
 
  • #3
ZaidAlyafey said:
See this .

For a. Can I just state that sin is bounded by -1 to 1 and therefore does not exist?

I went to the link you provided. I set bn=sin(1/n) and the limit does not equal 0, and therefore does not converge?

Also the same with b. Since the lim of bn which would be (1 +4/n)^n would also = 1 and not converge.

edit: reading further and discovered the next step.

Alright so by the divergence test as well I can confirm that both A and B diverge?
 
Last edited:
  • #4
ZaidAlyafey said:
Here you can use the comparison test .

For C I tried setting bn=1/(n^(2) -1) and took the limit to get that it goes to 0. Since bn is not greater than 0 an also converges.

Yes?
If so then I did the same thing for D and got the same answer.
 
Last edited:
  • #5
balloonhf said:
For a. Can I just state that sin is bounded by -1 to 1 and therefore does not exist?

I went to the link you provided. I set bn=sin(1/n) and the limit does not equal 0, and therefore does not converge?

Also the same with b. Since the lim of bn which would be (1 +4/n)^n would also = 1 and not converge.

edit: reading further and discovered the next step.

Alright so by the divergence test as well I can confirm that both A and B diverge?

Sorry, the first question is regarding sequences . So you test sin(1/n) as n goes to infinity , what is the limit ?
 
  • #6
balloonhf said:
For C I tried setting bn=1/(n^(2) -1) and took the limit to get that it goes to 0. Since bn is not greater than 0 an also converges.

Yes?
If so then I did the same thing for D and got the same answer.

\(\displaystyle \frac{\ln (n)}{n^2-1}< \frac{n}{n^2-1}\)

Sine the later converges , by comparison test the previous converges .
 

Related to Convergence and Divergence - Calculus 2

1. What is the difference between convergence and divergence in calculus 2?

Convergence and divergence are two important concepts in calculus 2 that describe the behavior of a sequence or a series. Convergence refers to the idea that a sequence or series approaches a specific value or limit as the number of terms increases. Divergence, on the other hand, means that the sequence or series does not approach a specific value and instead grows infinitely larger or smaller as the number of terms increases.

2. How can you determine if a sequence or series converges or diverges?

In calculus 2, there are several tests that can be used to determine convergence or divergence of a sequence or series. Some common tests include the comparison test, the ratio test, and the integral test. These tests involve analyzing the behavior of the terms in the sequence or series and can help determine if it converges or diverges.

3. What is the importance of understanding convergence and divergence in calculus 2?

Understanding convergence and divergence is crucial in calculus 2 as it allows us to determine the behavior of a sequence or series and the limits involved. This knowledge is essential in many applications of calculus, such as finding the sum of an infinite series or evaluating improper integrals.

4. Can a sequence or series be both convergent and divergent?

No, a sequence or series can only be either convergent or divergent. If it approaches a specific value or limit, it is considered convergent. If it does not approach a specific value, it is considered divergent.

5. How can convergence and divergence be applied in real-world situations?

Convergence and divergence have many real-world applications, such as in finance, physics, and engineering. For example, in finance, these concepts are used to determine the growth or decline of investments over time. In physics, they are used to analyze the behavior of physical systems, such as the motion of objects. In engineering, they are used to design and optimize systems and structures.

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