Converge, absolutely or conditionally?

  • Thread starter rcmango
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In summary, the given equation converges absolutely and also with the alternating series. The terms converge to zero, and the alternating series must approach 0 in order to converge. While sin(pi/anything) is not always equal to 0, sin(pi*n) is always 0 when n is an integer.
  • #1
rcmango
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Homework Statement



does it converge absolutely, converge conditionally, or diverge?

heres the equation: http://img219.imageshack.us/img219/4645/untitled29fy.jpg

Homework Equations



sin(pi/n)

The Attempt at a Solution



looks like the sin equation converges to zero.

i think this is an alternating series, but I'm not sure if this part converges or diverges?

thanks for any help here.
 
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  • #2
Yep, The terms converge to zero, and it converges absolutely, and therefore also with the alternating series.
 
  • #3
Great, thanks again.

..also, the alternating series must approach 0, in order to converge right?

..and, in this problem, sin(pi/anything) will always be 0, correct?
 
Last edited:
  • #4
Ahh no, if anything was a constant that it won't be zero.
 
  • #5
rcmango said:
..and, in this problem, sin(pi/anything) will always be 0, correct?

No ... but sin(pi*n) is always 0 when n is an integer. Maybe that's what you were thinking of (though it doesn't seem relevant to this question).
 

FAQ: Converge, absolutely or conditionally?

What is the difference between absolute convergence and conditional convergence?

Absolute convergence refers to a series that converges regardless of the order in which the terms are arranged, while conditional convergence refers to a series that only converges when the terms are arranged in a specific order.

How can you determine if a series converges absolutely or conditionally?

You can determine if a series converges absolutely or conditionally by using various convergence tests such as the ratio test, the root test, or the alternating series test. These tests can help determine the behavior of the series and whether it converges absolutely or conditionally.

What is an example of a series that converges absolutely?

An example of a series that converges absolutely is the geometric series, where each term is multiplied by a constant ratio. This series converges absolutely as long as the absolute value of the ratio is less than 1.

Are all convergent series also absolutely convergent?

No, not all convergent series are also absolutely convergent. Some series may converge conditionally, meaning they only converge when arranged in a specific order. This can be seen in the alternating harmonic series, which is conditionally convergent but not absolutely convergent.

What are some real-world applications of absolute and conditional convergence?

Absolute and conditional convergence have various applications in fields such as physics, engineering, and economics. For example, in physics, these concepts are used to calculate the behavior of electric and magnetic fields, while in economics, they are used to analyze the convergence of financial models. Additionally, understanding the convergence of series is important in the development of numerical methods for solving equations and simulations.

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