Is {q(n) * a(n)} = {p(n) * b(n)} (for all integer n's) a Cauchy Sequence?

In summary, the conversation discusses the proof that {q(n)}n (forall) Ns is a cauchy sequence, which is a sequence where the terms approach a limit and converge. The proof involves using the Maclaurin formula for e^x and showing that for any epsilon > 0, there exists a natural N such that the sum of 1/k! is less than epsilon. Additional hints are given, such as using the fact that 1/n! <= 1/2^(n) and showing that equivalent cauchy sequences multiplied together result in equivalent cauchy sequences.
  • #1
lainyg
3
0

Homework Statement



q(n) = Sum(from k=1 to n) 1/n!


Exercise 3: Prove that {q(n)}n(forall)Ns is a cauchy sequence.


Homework Equations



none.

The Attempt at a Solution



So many attempts at a solution. I know that a sequence is a cauchy sequence if for all epsilons greater than 0 there exists an N such that m,n >N and therefore the absolute value of q(m) minus (qn) is less than epsilon. A sequence is considered a cauchy sequence of its terms approach a limit (and converge). My problem is with proving this as it is a sum, and not letting it get messy with double factorials. How do I prove this?
 
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  • #2
I think you mean the sum of 1/k!, not 1/n!.

Hint 1: If [itex]\sum _{k = 1} ^{\infty}\frac{1}{k!}[/itex] converges, then for any [itex]\epsilon > 0[/itex], there exists a natural N such that [itex]\sum _{k=N} ^{\infty} \frac{1}{k!} < \epsilon[/itex]

Hint 2: What's the Taylor (or Maclaurin) expansion of ex?
 
  • #3
OK, so..

if qn converges, then for any epsilon>0 there exists a natural N such that (qn when N=k) is less than epsilon.

With the maclaurin formula we can write that e^x = the sum (from n=0 to infinity) of x^n/n!. Therefore can we just say that since the lim (as n approaches infinity) of q(n) is e, then it converges, and therefore is a cauchy sequence? Or do we still need to show that there's an N such that q(n) is less than epsilon (for any epsilon greater than 0)?
 
  • #4
I would guess that they are after a more direct proof than just saying 'I know it converges. Thus it is cauchy'. Would it help as a hint to note 1/n!<=1/2^(n) (at least for n>1)?
 
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  • #5
Another: Let {q(n)n} and {p(n)} (for all integer n's) be Cauchy Sequences which are equivalent. Further let {a(n)} and {b(n)} also be Cauchy Sequences which are equivalent.

Show {q(n) * a(n)} = {p(n) * b(n)} (for all integer n's)
 

1. What is a Cauchy sequence?

A Cauchy sequence is a sequence of numbers where the terms get closer and closer together as the sequence progresses. This means that the difference between any two terms in the sequence can be made arbitrarily small by choosing a large enough index.

2. What is the Cauchy sequence problem?

The Cauchy sequence problem is a mathematical problem that involves determining whether or not a given sequence is a Cauchy sequence. This problem is important in the field of analysis and is closely related to the concept of completeness in a metric space.

3. How is the Cauchy sequence problem solved?

The Cauchy sequence problem can be solved by using the Cauchy criterion, which states that a sequence is a Cauchy sequence if and only if for any positive number ε, there exists an index N such that for all n and m greater than or equal to N, the absolute value of the difference between the nth and mth term is less than ε.

4. What is the significance of the Cauchy sequence problem?

The Cauchy sequence problem is significant because it is closely related to the concept of completeness in a metric space. A metric space is complete if every Cauchy sequence in the space converges to a point in that space. This concept is important in many areas of mathematics, including analysis and topology.

5. What are some examples of Cauchy sequences?

Some examples of Cauchy sequences include the sequence of reciprocals, the sequence of decimal approximations to π, and the sequence of partial sums of a convergent series. It is important to note that not all sequences are Cauchy sequences, and it can be difficult to determine whether a given sequence is a Cauchy sequence or not.

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