Basic sequence help. (Convergence)

In summary, the conversation discusses the concept of convergence for a sequence and how to show that a sequence converges to zero. The definition of convergence is provided using \epsilon and N, and it is noted that this was likely introduced in a math class. The conversation also discusses how to use the definition to show that there exists an N for all n≥N such that -1 < nx(n) < 1.
  • #1
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Im struggling with the concept of this basic sequence question.

Let x(n) be a sequence such that lim(n->00) (nx(n)) = 0

i.e. it converges to zero...

How could i show that there is an N s.t. for all n≥N : -1 < nx(n) < 1

Any tips would be great.. I don't want an answer.. I want to be guided through it please.

Regards as ever.
Tam
 
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  • #2
What is the definition (using [itex]\epsilon[/itex] and N) of convergence for a sequence? Once you've written that, it should be straightforward to answer your question.
 
  • #3
the convergence of a seq:

A sequence {r^n}00n=0 converges if -1< r <1

Attempt at answer:

Same as above but replace 'r^n' with 'nx(n)

Thanks so far
 
  • #4
^ Wut?? That's the result of convergence of a geometric series, it's very different.

A sequence of real numbers [itex](x_n)[/itex] converges when there exists a number [itex]L \in \mathbb{R}[/itex] such that, for any [itex]\epsilon > 0[/itex], I can find a number [itex]N \in \mathbb{N}[/itex] so that [itex]|x_n - L| < \epsilon[/itex] whenever [itex]n > N[/itex]. This is very wordy but I'm sure this was introduced in your math class. The number L is the limit of the sequence [itex](x_n)[/itex].

What do you mean by "nx(n)"? Do you mean [itex]nx_n[/itex]?? If so, to show your desired result, use the definition from above. We know that for any number [itex]\epsilon > 0[/itex], I can find [itex]N \in \mathbb{N}[/itex] so that [itex]|nx_n - L| < \epsilon[/itex] whenever n > N. Fill in the details now: we know L = 0. Also, the previous statements work for any chosen [itex]\epsilon[/itex] -- how can you choose [itex]\epsilon[/itex] to get the result?
 
  • #5
Right:

|x(n) - L| < ε

therefore: |nx(n) - 0| < ε

therefore: nx(n) <

so, -ε < nx(n) < ε

so, take epsilon to be ε=1 and we have: -1 < nx(n) < 1

How does this look guys?

Thanks so far
 

1. What is the meaning of sequence convergence?

Sequence convergence refers to the behavior of a sequence where the terms in the sequence approach a specific value as the sequence progresses. This specific value is known as the limit of the sequence.

2. How do you determine if a sequence converges?

To determine if a sequence converges, you can use the limit test or the comparison test. The limit test involves finding the limit of the sequence and if it exists, the sequence converges. The comparison test compares the sequence to a known convergent or divergent sequence to determine its behavior.

3. Can a sequence converge to more than one value?

No, a sequence can only converge to one specific value, known as the limit. If a sequence converges to multiple values, it is considered to be divergent.

4. What is the difference between a convergent and a divergent sequence?

A convergent sequence approaches a specific value, known as the limit, as the sequence progresses. On the other hand, a divergent sequence does not have a specific limit and the terms in the sequence may grow infinitely or oscillate between different values.

5. How is sequence convergence related to series convergence?

Sequence convergence is a necessary condition for series convergence. If a sequence converges, the corresponding series will also converge. However, the converse is not true as a series can converge even if the corresponding sequence does not converge.

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