Is This Sequence a Cauchy Sequence?

In summary, the sequence (Sn) is defined as |Sn+1-Sn|<2-n and this is a Cauchy sequence as shown by using the hint and the polygon identity theorem.
  • #1
jaqueh
57
0

Homework Statement


Suppose the sequence (Sn) is defined as:
|Sn+1-Sn|<2-n
show that this is a cauchy sequence

Homework Equations


hint: prove the polygon identity such that
d(Sn,Sm)≤d(Sn,Sn+1)+d(Sn+1,Sn+2)...+d(Sm-1,Sm)

The Attempt at a Solution


I have defined Sm and Sn and created the inequality that:
d(Sn,Sm)=|2-n-2-m|>||Sn+1-Sn|-|Sm+1-Sm||
 
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  • #2
Using the hint, you'd get |Sn-Sm|<|Sn-Sn+1|+...+|Sm-1-Sm|<2-n+...+2-m
Is there any type of cauchy sequence (perhaps of partial sums) that would fit the latter half?
 
  • #3
ok I think I get what I can do now with the polygon ineq. except I thought that it would only be less than 2-n+...+2m-1also I don't know if I need to prove the polygon identity or not. if I did how would I go about it? would I use a fact of geometric series?
 
  • #4
Proving the "polygon" identity comes as a direct result of the definition of a metric, and the triangle inequality. A simple proof by induction would suffice.
While it is true you could remove the factor of 2-m from the inequality, you can add it on and still have the inequality hold true (plus it's better looking notation wise).
We know Ʃ2-n is a geometric series, and thus converges. Convergent sequences of partial sums (partial sums of a geometric series) are cauchy sequences. So Ʃ2-k from 1 to n minus Ʃ2-k from 1 to m would be less than ε
 

Related to Is This Sequence a Cauchy Sequence?

1. What is a Cauchy geometric sequence?

A Cauchy geometric sequence is a sequence of numbers in which each term is obtained by multiplying the previous term by a constant value, called the common ratio. It is named after the French mathematician Augustin-Louis Cauchy.

2. How can a Cauchy geometric sequence be proven to converge?

A Cauchy geometric sequence can be proven to converge by showing that the absolute value of the common ratio is less than 1. This ensures that the terms of the sequence approach 0 as n approaches infinity.

3. What is the formula for finding the sum of a Cauchy geometric sequence?

The formula for finding the sum of a Cauchy geometric sequence is S = a / (1 - r), where a is the first term and r is the common ratio. This is known as the infinite geometric series formula.

4. How does the proof of convergence for a Cauchy geometric sequence compare to other types of sequences?

The proof of convergence for a Cauchy geometric sequence is similar to other types of sequences, such as arithmetic and harmonic sequences. However, it requires an additional step of showing that the common ratio is less than 1.

5. Can a Cauchy geometric sequence diverge?

No, a Cauchy geometric sequence cannot diverge if the absolute value of the common ratio is less than 1. This is because as n approaches infinity, the terms of the sequence will approach 0, preventing the sequence from growing infinitely. However, if the absolute value of the common ratio is equal to or greater than 1, the sequence will diverge.

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