Hey there! Sign in to join this conversationNew here? Join for free
x Turn on thread page Beta

Choosing epsilon for a sequence that converges watch

    • Thread Starter
    Offline

    1
    ReputationRep:
    Problem + Solution
    Definition for convergence

    I'm really confused on how epsilon is chosen (i.e. the red arrows in the picture), and how those inequalities are worked out.

    In this example, is there a simpler value of epsilon that could be chosen?

    I (think I) understand how it works - you're proving that once you reach a certain point in the sequence (N) |an-l|, the difference between a term in the sequence and its limit, will always be smaller than an arbitrarily chosen small number.

    But I'm still lost on how epsilon is chosen.

    Many thanks.
    Offline

    20
    ReputationRep:
    ε can be any ( small number )... for instance if someone says ε = 0.05 then you put that in to the formula...

    so if N is greater than or equal to √(9/2*0.05) then |an - 0.5| < 0.05 as required
    • Thread Starter
    Offline

    1
    ReputationRep:
    (Original post by the bear)
    ε can be any ( small number )... for instance if someone says ε = 0.05 then you put that in to the formula...

    so if N is greater than or equal to √(9/2*0.05) then |an - 0.5| < 0.05 as required
    I think I understand all of that, but when you're solving it for yourself, how do you know to choose √(9/2ε)
    Offline

    20
    ReputationRep:
    (Original post by Enotdead)
    I think I understand all of that, but when you're solving it for yourself, how do you know to choose √(9/2ε)
    you just need to look at half a dozen worked problems to get the theme of finding epsilon
    • Thread Starter
    Offline

    1
    ReputationRep:
    (Original post by the bear)
    you just need to look at half a dozen worked problems to get the theme of finding epsilon
    ok, in this example is there anything else fairly obvious that epsilon could have been, or is this the only real solution?
    Offline

    20
    ReputationRep:
    (Original post by Enotdead)
    ok, in this example is there anything else fairly obvious that epsilon could have been, or is this the only real solution?
    i am good with that version :dontknow:
    Offline

    20
    ReputationRep:
    this may help you:

    https://bobobobo.wordpress.com/2008/...year-calculus/
    Offline

    17
    ReputationRep:
    (Original post by Enotdead)
    ok, in this example is there anything else fairly obvious that epsilon could have been, or is this the only real solution?
    You seem a little confused; your aim is find N given a particular value of \epsilon.

    So I suspect you mean is there anything else obvous that N could be (other than \sqrt{9/2\epsilon}.

    And the answer is yes. All you need to do is find an N that works (i.e.N such that for every n > N we have |a_n - a| &lt; \epsilon). So obviously if N works, anything bigger than N will work too.

    so \sqrt{100000/\epsilon} would work, for example.

    Note in particular that there's no requirement for N to be the smallest value that will work. It's important to understand this, since It's often impossibly hard to find the smallest value that will work.
    • Thread Starter
    Offline

    1
    ReputationRep:
    (Original post by DFranklin)
    .
    (Original post by the bear)
    .
    Thanks for the replies!
    edit: so many mistakes in this it was embarassing.
    Offline

    17
    ReputationRep:
    (Original post by Enotdead)
    edit wow I messed that one up - here's my uncertain-corrected solution, which I'm hoping some can check

    http://s21.postimg.org/8gxp59rfr/20151026_203856.jpg
    Your solution (both of them!) seem to make the claim that \sqrt{n+1} \leq 1/\sqrt{n+1}, which is clearly nonsense.

    This is a "fatal" mistake, there is nothing recoverable from what you've done after that point.
 
 
 
Reply
Submit reply
Turn on thread page Beta
Updated: October 26, 2015
Poll
Do I go to The Streets tomorrow night?
Useful resources

Make your revision easier

Maths

Maths Forum posting guidelines

Not sure where to post? Read the updated guidelines here

Equations

How to use LaTex

Writing equations the easy way

Student revising

Study habits of A* students

Top tips from students who have already aced their exams

Study Planner

Create your own Study Planner

Never miss a deadline again

Polling station sign

Thinking about a maths degree?

Chat with other maths applicants

Can you help? Study help unanswered threads

Groups associated with this forum:

View associated groups

The Student Room, Get Revising and Marked by Teachers are trading names of The Student Room Group Ltd.

Register Number: 04666380 (England and Wales), VAT No. 806 8067 22 Registered Office: International House, Queens Road, Brighton, BN1 3XE

Write a reply...
Reply
Hide
Reputation gems: You get these gems as you gain rep from other members for making good contributions and giving helpful advice.