Absolutely convergent/Conditionally convergent/Divergent

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In summary, the comparison test can be used to determine if a series converges or diverges, but it requires showing that the terms of the series are either larger or smaller than the corresponding terms of a known convergent or divergent series. The limit comparison test is another method that can be used, but it also requires finding a known convergent or divergent series that is either larger or smaller than the original series. Both methods can be complicated and may require more effort than directly proving the convergence or divergence of the original series.
  • #1
Neon32
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My question is can I use the comparison test? Since Cos is a bounded function?
 

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  • #2
Neon32 said:
View attachment 115148

My question is can I use the comparison test? Since Cos is a bounded function?
To use the Comparison Test, you have to have some idea about whether your series converges or diverges. If you believe your series diverges, you have to show that its terms are larger than the corresponding terms of the series you're comparing to. If you believe your series converges, you have to show that its terms are smaller than those of the series you're comparing to.
 
  • #3
If you are trying to check if the sum converges Keep in mind that this sum is not absolutely convergent, because
$$\sum_{n=1}^{\infty}\left |\frac{(-1)^n}{n} \right | = \sum_{n=1}^{\infty}\frac{1}{n}$$

and the harmonic series diverges

I think it would be easier to come up with another method

edit: I made a small reading mistake
 
  • #4
Mark44 said:
To use the Comparison Test, you have to have some idea about whether your series converges or diverges. If you believe your series diverges, you have to show that its terms are larger than the corresponding terms of the series you're comparing to. If you believe your series converges, you have to show that its terms are smaller than those of the series you're comparing to.

I'm a bit confused. Which one do you mean with "Your series"? The new series bn or the orignial series because it matters in this case. If you mean the orignial series.. then how I would know if it converges or not?
 
  • #5
MAGNIBORO said:
According to wikipedia the comparison test is for "infinite series with non-negative (real-valued) terms"
https://en.wikipedia.org/wiki/Direct_comparison_test

If you are trying to check if the sum converges Keep in mind that this sum is not absolutely convergent, because
$$\sum_{n=1}^{\infty}\left |\frac{(-1)^n}{n} \right | = \sum_{n=1}^{\infty}\frac{1}{n}$$

and the harmonic series diverges

This is a different way to solve it and I know. My question was about using the limit comparison test. I'm trying to solve it in different ways
 
  • #6
You can find convergent series that are larger / smaller for all elements, but that is complicated, and proving the convergence of those is more effort than proving the convergence of your original series.
 
  • #7
mfb said:
You can find convergent series that are larger / smaller for all elements, but that is complicated, and proving the convergence of those is more effort than proving the convergence of your original series.
Just to see if I understand it correctly. If an is the original series and bn is another series of my choice. If an<bn and bn is convergent then an is convergent as well
and if an>bn and bn is divergent then an is divergent? This is the theorem
 
  • #8
Neon32 said:
If an<bn and bn is convergent then an is convergent as well
That is not sufficient.
b_n=0, a_n=-1
It would be sufficient if a_n>0 for all n, but that is not the case here.

Same problem with the opposite direction.
 
  • #9
Neon32 said:
Which one do you mean with "Your series"?
The series you're working on-- the one in post #1.

Neon32 said:
My question was about using the limit comparison test.
You said comparison test, not limit comparison test, in post #1. My response was based on what you said.
 

1. What is the difference between absolutely convergent, conditionally convergent, and divergent series?

Absolutely convergent series are series in which the absolute values of each term in the series converge to a finite limit. Conditionally convergent series are series in which the terms themselves converge, but the absolute values of the terms do not. Divergent series are series in which neither the terms nor the absolute values of the terms converge.

2. How can I determine if a series is absolutely convergent, conditionally convergent, or divergent?

One way to determine the convergence of a series is by using the ratio test or the root test. If the limit of the ratio or root is less than 1, the series is absolutely convergent. If the limit is equal to 1, the series may be absolutely or conditionally convergent. If the limit is greater than 1, the series is divergent.

3. What are some real-world applications of absolutely convergent, conditionally convergent, and divergent series?

Absolutely convergent series can be used in calculus and engineering to calculate precise values, such as in Taylor series. Conditionally convergent series can be found in alternating current circuits and Fourier series. Divergent series can be used in economics and finance to model infinite growth or decay.

4. Can a series be both absolutely and conditionally convergent?

No, a series can only be either absolutely or conditionally convergent. If a series is absolutely convergent, it is automatically conditionally convergent. However, the inverse is not true.

5. What is the importance of understanding the convergence of series in mathematics?

Understanding the convergence of series is crucial for accurately calculating values and making predictions in various fields of mathematics and science. It also allows for the development of more complex mathematical concepts and theories, such as infinite series and integrals. Additionally, determining the convergence of series is important for identifying errors and ensuring the accuracy of calculations and models.

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