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Proving divergence of a sequence

  1. Sep 22, 2013 #1
    Please help me to solve following exercise (2.5.8) from Elementary Real Analysis by Thomson-Bruckner:

    Suppose that a sequence [itex]\{s_n\}[/itex] of positive numbers satisfies the condition [itex] s_{n+1} > \alpha s_n [/itex] for all ##n## where ##\alpha>1.## Show that ##s_n \to \infty.##

    I can't prove this using definition and given condition, I can only give an example of such sequence, ##\exp n## with ##\alpha = 2## but it seems useless for me. The only I know is that sequence is monotone and unbounded and so diverges.

    I'm only asking about a little hint that will give me a possibility to solve this by myself.

    Thanks a lot.
  2. jcsd
  3. Sep 22, 2013 #2


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    Hint: Does this sequence have a limit point?
  4. Sep 22, 2013 #3
    No. This sequence doesn't have a limit point but infinity. Right? But anyway I still don't understand. Let me to think some time. :-)
  5. Sep 22, 2013 #4


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    The condition implies sn > αns0 ->∞ as long as s0 > 0.
  6. Sep 22, 2013 #5
    Aha, now I understand the logic behind a given condition. Thank you!
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