• Support PF! Buy your school textbooks, materials and every day products Here!

Cauchy Sequence Problem

  • Thread starter lainyg
  • Start date
  • #1
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?
 

Answers and Replies

  • #2
AKG
Science Advisor
Homework Helper
2,565
3
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
3
0
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
Dick
Science Advisor
Homework Helper
26,258
618
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)?
 
Last edited:
  • #5
3
0
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)
 

Related Threads on Cauchy Sequence Problem

  • Last Post
Replies
11
Views
1K
  • Last Post
Replies
3
Views
956
  • Last Post
Replies
4
Views
1K
  • Last Post
Replies
1
Views
2K
  • Last Post
Replies
2
Views
2K
  • Last Post
Replies
1
Views
873
  • Last Post
Replies
10
Views
2K
  • Last Post
Replies
2
Views
670
  • Last Post
Replies
17
Views
3K
  • Last Post
Replies
9
Views
2K
Top