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Homework Help: Convergence of r^n

  1. Sep 25, 2011 #1
    I know this is like very basic, but my brain just somehow couldn't accept it!!

    1. The problem statement, all variables and given/known data
    I don't understand why does the sequence (rn) converges to 0 as n -> infinity when -1<|r|<1

    3. The attempt at a solution
    i did quite a few ways to convince myself.
    Firstly, we know that (1/r)n<(1/r) only if 0<r<1. So by squeeze theorm, (1/r)n converges to 0. Then it also holds for -1<r<0 cz |(1/r)n| = (1/r)n.

    this way seems to be correct but it doesn't seems to be convincing enough.
     
  2. jcsd
  3. Sep 25, 2011 #2

    CompuChip

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    Science Advisor
    Homework Helper

    If you want to make it more tangible, you could try it with some numbers:

    If r = 0.1, then you get { 0.1, 0.01, 0.001, 0.0001, ... } and you see this quickly tends to 0.
    If r = 0.5, then again { 0.5, 0.25, 0.125, 0.0625, ...}
    Even for r = 0.99, { 0.99, 0.9801, ... } goes much more slowly, but 0.99100 is already of the order of 5 x 10-5.

    Clearly, the boundary case is r = 1, as { 1, 1, 1, .... } never converges to 0.

    For the rigorous proof, refer to your own post :-) What you did there is correct.
     
  4. Sep 25, 2011 #3
    |r| < 1 so r = 1/x where |x|>1. r^n = 1/(x^n). If |x| >1 then clearly x^n becomes infinitely large as n goes to infinity. So 1/(big number) goes to zero.
     
  5. Sep 25, 2011 #4
    Ok. Thanks so much
     
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