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Limit of a sequence

  1. Feb 7, 2012 #1
    1. The problem statement, all variables and given/known data
    Let [x] be the greatest integer ≤x. For example [itex] [\pi ]=3 [/itex]
    and [3]=3
    Find [itex] lim a_n [/itex] and prove it.
    a) [itex] a_n=[\frac{1}{n}] [/itex]
    b) [itex] a_n=[\frac{10+n}{2n}] [/itex]
    3. The attempt at a solution
    for the first one it will converge to zero.
    so can I write [itex] \frac{1}{n}< \epsilon [/itex]
    then I can just pick an n large enough to make that work.
    for part b, it also looks like it will converge to zero, but a little slower.
    so [itex] \frac{10+n}{2n}< \epsilon [/itex] and then solve for n in terms of ε.
    Or is there something I am not taking into account with the greatest integer deal?
     
  2. jcsd
  3. Feb 7, 2012 #2

    Deveno

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    Science Advisor

    for the first one:

    you are NOT trying to "solve for ε in terms of n", but rather, the other way around. you need to find a (possibly large) value for N so that n > N makes 1/n "small" (less than epsilon). epsilon is arbitrary, but assumed as given.

    for the second one, i would note that:

    (n+10)/2n = 1/2 + 5/n.

    if you find an N such that n > N means 5/n < 1/2,

    wouldn't [(n+10)/2n] = 0 for all such n?

    it seems to me the floor function "speeds up" the convergence, instead of making it slower.
     
  4. Feb 7, 2012 #3
    thanks for your help, ok
    so on the second one [itex] .5+\frac{5}{n} < \epsilon [/itex]
    so then I just pick an n large enough to make it less than epsilon.
     
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