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Lim inf and Lim sup

  1. Oct 10, 2009 #1
    1. The problem statement, all variables and given/known data

    Given the sequence, 1/2 + 1/3 + 1/(2^(2)) + 1/(3^(2)) + 1/(2^(3)) + 1/(3^(3)) + ...., Describe the terms of the sequence and use it to compute the lim inf (a_n+1)/(a_n); lim sup (a_n+1)/(a_n); lim inf (a_n)^(1/n); lim sup (a_n)^(1/n).

    2. Relevant equations


    3. The attempt at a solution

    First, I found the formula for the sequence, which is [tex]\Sigma[/tex] i=1 to infinity of [1/(2^(i)) + 1/(3^(i))].

    I wrote out some terms of the sequence, but I'm having a hard time pulling out a subsequence to compute the ratio and root tests.
     
  2. jcsd
  3. Oct 10, 2009 #2

    Mark44

    Staff: Mentor

    First off, 1/2 + 1/3 + 1/(2^(2)) + 1/(3^(2)) + 1/(2^(3)) + 1/(3^(3)) + ... is an infinite sum (AKA infinite series), not a sequence. Every infinite sum has a sequence associated with it, the sequence of partial sums, {Sn}, which is defined this way:
    [tex]S_n~=~\sum_{i = 1}^n a_n[/tex]

    Is an defined for your infinite sum? If not, it would be more convenient to define the general term in your series as an = 1/2n + 1/3n. If an is defined in this way or can be defined this way, the ratio an + 1/an is fairly easy to calculate, and comes out to (3n+1 + 2n+1)/(6(3n + 2n)). You should be able to work with this to get your lim sup and lim inf values

    On the other hand, if an has a different formula for the even terms and the odd terms, it's going to be more difficult to calculate the aforementioned ratio.
     
  4. Oct 10, 2009 #3
    The second part of the problem says I should consider both cases, positive and negative, when computing the lim inf and lim sup. I've never seen this type of problem before and I'm totally sure what it is asking me to do.
     
  5. Oct 10, 2009 #4

    Mark44

    Staff: Mentor

    You didn't include that information in your problem description. What do you mean, both cases, positive and negative?
     
  6. Oct 10, 2009 #5
    Yeah, both cases positive and negative.
     
  7. Oct 10, 2009 #6

    Mark44

    Staff: Mentor

    Again, what do you mean by the positive and negative cases?
     
  8. Oct 10, 2009 #7
    From my notes: After writing down a formula for {a_n}, describe {a_n} when n is even (n=2k) and when n is odd (2k-1).
     
    Last edited: Oct 10, 2009
  9. Oct 10, 2009 #8

    Mark44

    Staff: Mentor

    That information would have answered my questions in post #2. This doesn't have anything to do with positive and negative, but rather, odd and even terms.

    So finally, we're getting somewhere.

    Here are some more questions.
    What do the a2k terms of this series look like?
    What do the a2k - 1 terms look like?

    If these are too hard, what do a2, a4, a6, and so on look like? Can you generalize this?

    What do a1, a3, a5, and so on look like? Can you generalize this?
     
  10. Oct 11, 2009 #9
    How are you getting the lim inf and lim sup vales out of (3n+1 + 2n+1)/(6(3n + 2n))?
     
  11. Oct 11, 2009 #10

    Mark44

    Staff: Mentor

    I'm not. Look at post #8 again, and answer the questions I've asked.
     
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