Searching for a particular kind of convergent sequence

In summary, the conversation is about finding an example of a complex sequence (x_n) that converges to 0 but is not in ℓ^p for any p≥1, meaning the series ∑|x_n|^p is never convergent. The suggested example is 1/log(n+1) and it is suspected to work, but the estimates needed to demonstrate divergence are still being figured out. The person requesting the example is having trouble establishing the divergence and asks for more explicit explanation. Another person suggests that (1/log(n))p > 1/n for sufficiently large n could demonstrate divergence.
  • #1
A-ManESL
8
0
I want an example of a complex sequence [tex](x_n)[/tex] which converges to 0 but is not in [tex]^p[/tex], for [tex]p\ge 1[/tex] i.e. the series [tex]\sum |x_n|^p[/tex] is never convergent for any [tex]p\ge 1[/tex]. Can someone provide an example please?
 
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  • #2
I suspect 1/log(n+1) will work, but I haven't checked divergence for p > 1.

*EDIT* I'm fairly certain it works. I'll let you figure out the estimates needed to demonstrate divergence (hint: you don't need obscure series tests).
 
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  • #3
I am having trouble establishing the divergence. Can you be more explicit? Thanks.
 
  • #4
For divergence I would guess (1/log(n))p > 1/n for sufficiently large n.
 

Related to Searching for a particular kind of convergent sequence

1. What is a convergent sequence?

A convergent sequence is a sequence of numbers that approaches a specific value as the index approaches infinity. In other words, as the sequence continues, the terms get closer and closer to a single limit.

2. How do you search for a particular kind of convergent sequence?

In order to search for a particular kind of convergent sequence, you must first determine what type of sequence you are looking for (i.e. arithmetic, geometric, etc.). Then, you can use various mathematical techniques such as algebraic manipulation or graphing to find the specific sequence that meets your desired criteria.

3. What are the properties of a convergent sequence?

A convergent sequence has a single limit, meaning that all terms in the sequence approach the same value as the index increases. Additionally, the terms in a convergent sequence must get closer and closer to the limit as the index increases.

4. Can a convergent sequence have a limit of infinity?

No, a convergent sequence cannot have a limit of infinity. The limit of a convergent sequence must be a real number, meaning that the terms in the sequence must approach a specific value as the index increases.

5. How is a convergent sequence different from a divergent sequence?

A convergent sequence approaches a specific value as the index increases, while a divergent sequence does not have a single limit and the terms may either increase or decrease without approaching a specific value. Additionally, a convergent sequence must have terms that get closer and closer to the limit, while a divergent sequence may have terms that increase or decrease without any pattern.

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