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Sequence Convergence

  1. Dec 2, 2009 #1
    1. The problem statement, all variables and given/known data

    Determine if the sequence an converges.

    an = (13(4n)+11)/(10(5n))

    2. Relevant equations

    N/A

    3. The attempt at a solution

    [tex]\lim_{n \to \infty} (\frac{(3(4^n))+11}{10(5^n)})[/tex]
    = [tex]1/10 \lim_{n \to \infty} (\frac{(3(4^n))+11}{5^n})[/tex]
    = [tex]1/10 \lim_{n \to \infty} (\frac{(3(4^n)}{5^{n}}) + \lim_{n \to \infty} (\frac{11}{5^n}) [/tex]

    I feel like I am missing something very basic. Thank you for your help!
     
    Last edited: Dec 2, 2009
  2. jcsd
  3. Dec 2, 2009 #2

    lanedance

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    how did you get your last line? where did 52 come from?

    what is [tex] \lim_{n \to \infty} (\frac{4}{5})^n[/tex]?
     
  4. Dec 2, 2009 #3
    Oops! That was a typo -- fixed. Sorry about that.

    And, if I write out the sequence, I would have to say the limit of (4/5)^n = 0.
    Am I allowed to split the limit like I did, and, if so, are the limits separately 0 and 0, so that the final answer is "converges to zero"?
     
  5. Dec 2, 2009 #4

    lanedance

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    it depends what has been proved, but where the limie exists and is finite for both f & g, generally it is fine to assume
    [tex]\lim_{n \to \infty} (f(n) +g(n)) = \lim_{n \to \infty} f(n) +\lim_{n \to \infty}g(n)[/[/tex]

    [tex]\lim_{n \to \infty} (\frac{(3(4^n))+11}{10(5^n)}) = \lim_{n \to \infty} \frac{3}{10}(\frac{4}{5})^n + \lim_{n \to \infty} \frac{11}{10}(\frac{1}{5})^n[/tex]
     
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