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Confirm limit of this sequence by using definition of limit

  1. Aug 14, 2014 #1

    s3a

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    1. The problem statement, all variables and given/known data
    I'm referring to the question and solution for part (b) in the attached TheProblemAndSolution.jpg file.

    2. Relevant equations
    Definition of limit.

    3. The attempt at a solution
    Should the equation with the two things in brackets have absolute value bars instead of brackets?

    Also, according to this Wikipedia link ( http://en.wikipedia.org/wiki/Limit_of_a_sequence#Formal_Definition ), N should be a natural number and ϵ should be a real number but the solution in my book gives N as a function of ϵ, which means that it could be the case that N is also real and non-natural (such as an irrational number for example). Is there a contradiction here, or is there something I'm missing?

    Any input would be GREATLY appreciated!
     

    Attached Files:

  2. jcsd
  3. Aug 14, 2014 #2

    mfb

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    Yes.

    You can simply round up. If something is true for all n larger than 245.3, then it is also true for all n larger than 246.
     
  4. Aug 14, 2014 #3

    s3a

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    Sorry, I double-posted.
     
  5. Aug 14, 2014 #4

    s3a

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    Thanks for confirming.

    Thanks for this answer too, but there is a bit more I need to understand.

    Is it more strict than neccesary (for that Wikipedia link I gave) to require that n and N be natural numbers?

    Shouldn't N be allowed to be any real number, and shouldn't n be allowed to be any integer (including negative ones)?
     
  6. Aug 15, 2014 #5

    s3a

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    Actually, I have a feeling it just doesn't matter if N is a natural number or not, but I wanted to know if there are varying definitions (among different textbooks and websites, etc), or if every or most sources have the definition that the Wikipedia link uses.
     
  7. Aug 15, 2014 #6

    s3a

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    Okay, so, if anyone else cares, I got the answer to my problem.:

    Since there is no regulatory body for standardizing definitions in mathematics, not all definitions of the limit of a sequence are exactly the same, but they are all largely similar, so in the case of the problem in this thread, I could use the definition where N is a real number (instead of rounding up to the nearest natural number).

    Also, apparently, indices for terms of sequences are always natural numbers.

    Also, thanks, mfb.
     
  8. Aug 15, 2014 #7

    Fredrik

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    There's nothing wrong with a definition that says that N is a positive real number, but it's a bit weird. What the standard definition of the limit of a sequence tells you is this: A real number x is a limit of a sequence S if and only if every open interval centered at x contains all but a finite number of terms of the sequence. The ##\varepsilon##-N statement of the definition is just a way to rephrase that in a way that makes it perfectly clear what "all but a finite number" means. So when you want to prove that ##x_n\to x##, the idea is to identify a specific term ##x_N## such that it, and all subsequent terms, are in the given interval.

    N must depend on ##\varepsilon##, because ##\varepsilon## tells you the size of the interval. For example, consider the sequence 1/n and an open interval centered at 0. The smaller the interval, the further you have to go into the sequence to find a term such that all subsequent terms are also in that interval.

    Yes, the indices are always integers, in the sense that if they're not, we would call it a "net" instead of a "sequence". A sequence can be defined as a function from the set of natural numbers into some other set. A net is defined as a function from a directed set into some other set. You don't have to learn about nets or directed sets in this course. I just thought it might be interesting to know.
     
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