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Convergence of a series

  1. Jan 30, 2010 #1
    Hey guys,
    1. The problem statement, all variables and given/known data
    Prove: If for every n [tex] a_{n}>0 [/tex] and [tex] \frac{a_{n+1}}{a_{n}}<1 [/tex] then the series [tex]lim_{n->\infty} a_{n}<0 [/tex]



    3. The attempt at a solution
    We know that [tex] a_{n} [/tex] is lowerly bounded by 0 and upwardly bounded by [tex] a_{1} [/tex]. we also know that it is monotonic and decreasing and so congerges. But how do I show that it converges to 0. What is to stop it from converging to, say, .5?
    Thanks
    Tal
     
    Last edited: Jan 30, 2010
  2. jcsd
  3. Jan 30, 2010 #2
    When you say series, are you referring to a summation or a sequence?

    If an < 0 for all n, then all your terms are negative. So apply this fact to an+1/an < 1
     
  4. Jan 30, 2010 #3
    Sorry, I made a typo. that was an>0.
     
  5. Jan 30, 2010 #4
    and I am refering to a sequence, not a summation. Pardon me, english is not my native language.
     
  6. Jan 30, 2010 #5
    Ahh, I misread the question. it was prove or disprove. I found a counter example.
    Thanks anyway.
    Tal
     
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