Proving divergence of a sequence

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    Divergence Sequence
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Discussion Overview

The discussion revolves around proving the divergence of a sequence of positive numbers under a specific condition, where the sequence satisfies \( s_{n+1} > \alpha s_n \) for all \( n \) with \( \alpha > 1 \). Participants explore the implications of this condition and seek hints for a proof, focusing on the concepts of limits and monotonicity.

Discussion Character

  • Homework-related
  • Exploratory
  • Mathematical reasoning

Main Points Raised

  • One participant presents a specific exercise from a textbook and expresses difficulty in proving that the sequence diverges to infinity using the given condition.
  • Another participant suggests considering whether the sequence has a limit point.
  • A participant responds that the sequence does not have a limit point, suggesting that it approaches infinity instead.
  • Another contribution indicates that the condition implies \( s_n > \alpha^n s_0 \), leading to the conclusion that \( s_n \to \infty \) as long as \( s_0 > 0 \).
  • A later reply indicates a realization of the logic behind the condition, suggesting some understanding has been achieved.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the proof itself, but there is a shared understanding that the sequence diverges to infinity based on the given condition.

Contextual Notes

The discussion does not resolve the specific steps needed to construct a formal proof, and participants express varying levels of understanding regarding the implications of the condition.

dimanet
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Hello!
Please help me to solve following exercise (2.5.8) from Elementary Real Analysis by Thomson-Bruckner:

Suppose that a sequence \{s_n\} of positive numbers satisfies the condition s_{n+1} > \alpha s_n 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.
 
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Hint: Does this sequence have a limit point?
 
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.
 
The condition implies sn > αns0 ->∞ as long as s0 > 0.
 
Aha, now I understand the logic behind a given condition. Thank you!
 

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