Adsolute and conditional convergence of alternating series

In summary, the conversation discusses the convergence of alternating series and the use of Leibniz's theorem in proving conditional convergence. It is mentioned that the series for tan(pi/n) diverges but can be shown to converge conditionally for n>3 using the alternating series test. There is also a discussion on whether the series is undefined or divergent for n>0. Ultimately, it is clarified that conditional convergence means the series converges but taking the absolute value would result in divergence.
  • #1
blursotong
15
0
i have a question regarding adsolute and conditional convergence of alternating series.

- i know that summation of [ tan(pi/n) ] diverge, but how do we proof it converge conditionally? (ie, (-1)^n tan(pi/n) ]

can Leibiniz's theorem be used in this case? but tan(pi/2) is infinite?

any help is appreciated. =D
 
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  • #2
If you write it as sum n=3 to infinity then you can use the alternating series test. If the series includes n=2 then it would be undefined.
 
  • #3
so alternating series of tan(pi/n) converge conditionally for n>3 only ?
for n>0 it is diverge?
 
  • #4
Conditional convergence of an alternating series means that it converges but if you take the absolute value it diverges?
 
  • #5
blursotong said:
so alternating series of tan(pi/n) converge conditionally for n>3 only ?
for n>0 it is diverge?

Maybe. Read the fine print in the definition and consult a lawyer. I would prefer to call the case n>0 undefined rather than divergent.
 
  • #6
dacruick said:
Conditional convergence of an alternating series means that it converges but if you take the absolute value it diverges?

Well, yes.
 

1. What is the definition of absolute convergence for an alternating series?

A series is said to be absolutely convergent if the sum of the absolute values of its terms converges, regardless of the order in which the terms are added.

2. How is conditional convergence different from absolute convergence?

Conditional convergence occurs when a series is convergent, but not absolutely convergent. In other words, the sum of the terms of the series converges, but the sum of the absolute values of the terms does not.

3. What is the alternating series test and how is it used to determine convergence?

The alternating series test is a method used to determine the convergence of an alternating series. It states that if the terms of an alternating series decrease in absolute value and ultimately approach zero, then the series converges.

4. Can an alternating series be both absolutely and conditionally convergent?

No, an alternating series can only be either absolutely convergent or conditionally convergent, but not both.

5. What are some real-world applications of alternating series convergence?

Alternating series convergence can be used in various fields such as physics, engineering, and economics to model and analyze real-world phenomena. For example, in physics, it can be used to calculate the movement of pendulums or the behavior of alternating electrical currents. In economics, it can be used to analyze the fluctuation of stock prices or interest rates over time.

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