Question on limits of alt. signs

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In summary, if you are taking the limit of a sequence of numbers that is An and An alternates in signs, you should first try to redefine n as only an even number that is of the form n=2k, where k is an integer. If the limit for n even and for n odd is the same than the series converges but if the limit as n is odd and as n is even are not the same, then the sequence does not have a limit, so it diverges.
  • #1
nuclearrape66
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If I'm taking the limit of a sequence of numbers that is {An} and An alternats in signs.
am i suppose to take the absolute value to determine the limit. and if the absolute value of An diverges then the lmit diverges?

or how am i suupose to take the limit of alternating signs?
 
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  • #2
well, the sequence probbably alternate its sign as you plug in an even or an odd number. So what u need to do is try first to redefine n as only an even number that is of the form n=2k, where k is an integer, and the other time define n to be only odd, that is n=2k-1. an look what happenes when you take the limit? If the limit for n even and for n odd is the same than the series converges but if the limit as n is odd and as n is even are not the same, then the sequence does not have a limit, so it diverges.
Do you understand what you need to do?
 
  • #3
so youre saying to plug in 2n for n and get its limit...


then plug in 2n + 1 for n and find its limit...

and if they both go towards the same thing then it converges to that number?
 
  • #4
well, you can let n=2k, and then as n-->infinity, also k-->infinity, the same with
n=2k-1, as n-->infinity, also k--->infinity? Then if the both limits go to the same number the overall limit of that sequence will be that particular number, in contrary the limit does not exist.
 
  • #5
so for (-1)^n(1/n) i would do:

lim (-1)^2n(1/2n) = 0

and

lim (-1)^2n+1(1/2n + 1 ) = 0

and therefore it's 0.

but what about (-1)^n(n/n+2)
 
  • #6
well the last one does not have a limit, since when n:=2n, and when n:=2n+1, you will get
1, and -1 respectively.
 
  • #7
oh i see, so so then if you do the absolute value thing, where you take the limit of any alternating sequeence and if that diverges then the alt sequence diverges if it converges then the sequence converges.

does that work all the time for these?
 

Related to Question on limits of alt. signs

1. What is the significance of alternative signs?

Alternative signs are used to represent a concept or idea in a different way than traditional symbols or words. They offer a different perspective and can help to convey complex ideas more effectively.

2. How do alternative signs compare to traditional signs?

Alternative signs often have a more abstract or creative approach compared to traditional signs, which tend to be more practical and straightforward. Alternative signs may also be less widely recognized or understood.

3. What are the limitations of alternative signs?

One limitation of alternative signs is that they may not be universally understood or recognized, which can limit their effectiveness in communication. They may also be less regulated and standardized compared to traditional signs, leading to potential confusion or misinterpretation.

4. How can alternative signs be used in scientific research?

Alternative signs can be used in scientific research to represent complex concepts or data in a simplified and visually appealing manner. They can also be used to add a creative element to research presentations or publications.

5. Are there any ethical considerations when using alternative signs?

It is important to consider the potential impact of alternative signs on different audiences, including those with visual impairments or different cultural backgrounds. The use of alternative signs should also be accompanied by clear explanations to ensure understanding and avoid misinterpretation.

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