A community for students.

Here's the question you clicked on:

55 members online
  • 0 replying
  • 0 viewing

anonymous

  • 5 years ago

how can we prove that a(n) = n/n+1 converges by using the definition? I know how to find the limit, yet I can't seem to find a way to prove it using the definition

  • This Question is Closed
  1. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    I might be able to help you. I would consider the following:\[\lim_{n->\infty}\frac{n}{n+1}=\lim_{n->\infty}\frac{1}{1+\frac{1}{n}}\]and from limit laws, \[\frac{1}{1+\lim_{n \rightarrow \infty} \frac{1}{n}}\]Now, for \[\lim_{n \rightarrow \infty}\frac{1}{n}\]choose \[\epsilon >0\] small. Let N be the first integer greater than \[\frac{1}{\epsilon}\]Then\[\forall n>N \rightarrow n>\frac{1}{\epsilon} \iff \forall n>N \rightarrow \epsilon >\frac{1}{n} \iff \frac{1}{n}< \epsilon\]Since n>0,\[0<\frac{1}{n}< \epsilon \iff 0<\frac{1}{n} - 0 < \epsilon\] and so, by definition of the limit,\[\frac{1}{n} \rightarrow 0 \] as \[n \rightarrow \infty\]

  2. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    So,\[\lim_{n->\infty}\frac{n}{1+n}=\lim_{n->\infty}\frac{1}{1+\frac{1}{n}}=\frac{1}{1+\lim_{n->\infty}\frac{1}{n}}=\frac{1}{1+0}=1\]

  3. nowhereman
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    You even do not need that limit law, if you already know the limit. Just calculate \[|1-\frac{n}{1+n}| = \frac{1}{1+n}\] so need only find n for ε so that \[\frac 1 {1+n} < ε ⇔ n > \frac 1 ε - 1\] Thus if you choose \[ n = \frac 1 ε \] everythings fine.

  4. anonymous
    • 5 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    thank you :)

  5. Not the answer you are looking for?
    Search for more explanations.

    • Attachments:

Ask your own question

Sign Up
Find more explanations on OpenStudy
Privacy Policy

Your question is ready. Sign up for free to start getting answers.

spraguer (Moderator)
5 → View Detailed Profile

is replying to Can someone tell me what button the professor is hitting...

23

  • Teamwork 19 Teammate
  • Problem Solving 19 Hero
  • You have blocked this person.
  • ✔ You're a fan Checking fan status...

Thanks for being so helpful in mathematics. If you are getting quality help, make sure you spread the word about OpenStudy.

This is the testimonial you wrote.
You haven't written a testimonial for Owlfred.