Proving the Cauchy Sequence of (An)

In summary, the question is how to prove that the sequence (An) = (1+(1/sqrt(2))+(1/Sqrt(3))+...+(1/sqrt(n))-2sqrt(n))) is Cauchy, and the suggested method is to use the monotone convergence theorem and create a lower bound using the function f(x) = 1/sqrt(x). The conversation also mentions looking for alternative methods.
  • #1
hypermonkey2
102
0
How could i show that the sequence
(An)= (1+(1/sqrt(2))+(1/Sqrt(3))+...+(1/sqrt(n))-2sqrt(n))) is Cauchy?
Thanks in advance!
 
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  • #2
hypermonkey2 said:
How could i show that the sequence
(An)= (1+(1/sqrt(2))+(1/Sqrt(3))+...+(1/sqrt(n))-2sqrt(n))) is Cauchy?
Thanks in advance!

You need to show it is convergent.
Define [tex]f(x) = \frac{1}{\sqrt{x}}[/tex]. Now confirm that [tex]\sum_{j=1}^{n} \frac{1}{\sqrt{j}} \geq \int_1^n \frac{1}{\sqrt{x}}dx[/tex]. Use this to create a lower bound. Now apply the monotone theorem.
 
  • #3
Hmm.. I was hoping that there would be a way to do this straight from the definition of a Cauchy sequence, without use of the notion of a definite integral.. thanks though! any other ideas?
 

1. What is a Cauchy sequence?

A Cauchy sequence is a sequence of numbers where the terms become arbitrarily close to each other as the sequence progresses. This means that for any small distance, there is a point in the sequence after which all the terms are within that distance from each other.

2. How do you prove that a sequence is Cauchy?

To prove that a sequence is Cauchy, you need to show that for any small distance, there exists a point in the sequence after which all the terms are within that distance from each other. This can be done by using the definition of a Cauchy sequence and showing that it holds true for the given sequence.

3. What is the importance of proving the Cauchy sequence?

Proving that a sequence is Cauchy is important in mathematics as it guarantees that the sequence converges to a limit. This is useful in various areas of mathematics, such as in the convergence of infinite series and in the definition of continuity in analysis.

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

A Cauchy sequence is a sequence where the terms become arbitrarily close to each other, while a convergent sequence is a sequence that has a limit. All convergent sequences are Cauchy, but not all Cauchy sequences are convergent.

5. Can a sequence be Cauchy but not convergent?

Yes, a sequence can be Cauchy but not convergent. This means that the terms of the sequence become arbitrarily close to each other, but the sequence does not have a limit. This can happen when the sequence oscillates between two values or when the terms become closer and closer to each other, but do not approach a specific value.

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