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Prove the sequence converges

  1. Sep 21, 2015 #1
    Hello Physics Forums community! I've been struggling for a while with this one emoticon-0121-angry.gif ; so basically a sequence a_n is given to us such that the sequence b_n defined by b_n = pa_n + qa_(n+1) is convergent where abs(p)<q. I need to prove a_n is convergent also. Any hint would be of so much help, thank you.

    (I've tried proving it is Cauchy but no insight ever comes, just messing with inequalities. I haven't fully understood the role played by abs(p)<q)
     
  2. jcsd
  3. Sep 21, 2015 #2

    Dick

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    You can start by trying to think what could go wrong if ##|p|<q## isn't true. Suppose ##a_n=(-1)^n## and ##p=1## and ##q=1##.
     
  4. Sep 21, 2015 #3
    I'm well aware of the counter examples when the inequality doesn't hold. I just can't find a way to prove the theorem; and to do this I need to fully understand the importance of the inequality. I can't see it intuitively
     
  5. Sep 22, 2015 #4
    You can use the definition in terms of epsilon.
     
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