Proving divergence of a sequence

  • Thread starter dimanet
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  • #1
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Main Question or Discussion Point

Hello!
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.
 

Answers and Replies

  • #2
UltrafastPED
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Hint: Does this sequence have a limit point?
 
  • #3
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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. :-)
Thanks.
 
  • #4
mathman
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The condition implies sn > αns0 ->∞ as long as s0 > 0.
 
  • #5
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Aha, now I understand the logic behind a given condition. Thank you!
 

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