Proving divergence of a sequence

In summary, the conversation is about solving an exercise from a book on Elementary Real Analysis. The exercise asks to show that a sequence of positive numbers satisfying the condition sn+1 > αsn for all n, where α > 1, will diverge. The person asking for help is only able to provide an example of such a sequence but is looking for a hint to solve it on their own. The hint given is that the sequence does not have a limit point, which implies that it will diverge. After some thinking, the person understands the logic behind the given condition.
  • #1
3
0
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.
 
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  • #2
Hint: Does this sequence have a limit point?
 
  • #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. :-)
Thanks.
 
  • #4
The condition implies sn > αns0 ->∞ as long as s0 > 0.
 
  • #5
Aha, now I understand the logic behind a given condition. Thank you!
 

1. What does it mean for a sequence to diverge?

A sequence diverges if its terms become increasingly larger or smaller without approaching a finite limit. In other words, the terms of the sequence do not converge towards a specific value.

2. How can you prove that a sequence diverges?

To prove divergence of a sequence, you can use the definition of divergence, which states that for any positive real number M, there exists a positive integer N such that for all n greater than N, the absolute value of the nth term of the sequence is greater than M.

3. Can a sequence diverge to infinity?

Yes, a sequence can diverge to infinity if its terms become arbitrarily large as n increases. This is known as unbounded divergence.

4. Are there different types of divergence?

Yes, there are two types of divergence: unbounded divergence and oscillating divergence. Unbounded divergence occurs when the terms of the sequence become increasingly larger or smaller without limit, while oscillating divergence occurs when the terms of the sequence alternate between two or more values without limit.

5. How is divergence related to the concept of limits?

Divergence is the opposite of convergence, which is the behavior of a sequence that approaches a specific value or limit. While a convergent sequence approaches a limit, a divergent sequence does not have a limit. In other words, a sequence can only converge or diverge, there is no in-between.

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