Turn on thread page Beta
    • Thread Starter
    Offline

    0
    ReputationRep:
    Hi, i need help with the following questions but i need the answers explained if possible.

    4.(c) Is (Z4 - { [0]4}, \odot, [1]4) a group?

    5.(a) Give a precise definition of what it means to say that (an) converges to l as n tends to infinity.

    The answer is below,
    \forall\epsilon>0, \exists N \varepsilon N such that \forall n > N, |an - l| < \epsilon

    N = natural numbers
    (b) By directly applying the definition, prove that,
    an : = (√n)/(1-3√n)
    converges to a limit, that you are expected to determine.

    7. Determine, with jusification, for which of the following the sequence (an) converges; where a limit exists determine it.
    (b) an : = (2^n)×(n²)

    8. Determine with justification, which of the following series converge.
    (c) ∑ n=1 to n=∞ (n/n+1)
    You may appeal to the general limit theorems.

    Thanks a lot
    Offline

    15
    ReputationRep:
    5.(a) Give a precise definition of what it means to say that (an) converges to l as n tends to infinity.
    For every \epsilon &gt;0 there exists N_{\epsilon} (note the subscript \epsilon shows the dependence of N on \epsilon) such that |a_{n}-l|&lt;\epsilon for all n&gt;N
    This is a definition and probably just needs to be learnt. You can see from the definition that it's kind of obvious though - as N_{\epsilon} increases a_{n} gets closer and closer to l.

    N = natural numbers
    (b) By directly applying the definition, prove that,
    an : = (√n)/(1-3√n)
    converges to a limit, that you are expected to determine.
    Before applying the definition we need an idea of what the limit is.
    We can informally write a_{n}=\frac{1}{\frac{1}{\sqrt{n}  }-3} and so guess that the limit is \frac{-1}{3}
    Then:
    |a_{n}+\frac{1}{3}|=|\frac{\sqrt  {n}}{1-3\sqrt{n}}+\frac{1}{3}| \\

=|\frac{3\sqrt{n}+1-3\sqrt{n}}{3(1-3\sqrt{n})}| \\

=|\frac{1}{3(1-3\sqrt{n})}| \\

&lt;|\frac{1}{1-3\sqrt{n}}| \\

&lt;\frac{1}{\sqrt{n}} (n&gt;1)\\

&lt;\frac{1}{n} \\

&lt;\epsilon \forall n&gt;N=\frac{1}{\epsilon}
    Offline

    15
    ReputationRep:
    7. Determine, with jusification, for which of the following the sequence (an) converges; where a limit exists determine it.
    (b) an : = (2^n)×(n²)
    We can see 'casually' that 2^{n}.n^2 \rightarrow \infty as n\rightarrow \infty so we would perhaps be justified to disprove convergence by applying the definition for a_{n} \rightarrow \infty as n \rightarrow \infty which is a_{n}&gt;A for every n&gt;N
    a_{n}=2^{n}.n^2 &gt; 2^{n} (n&gt;1) \\

&gt;A \forall n&gt;N=log_{2}A
    Hence a_{n} \rightarrow \infty as n\rightarrow \infty and a_{n} is not convergent.

    8. Determine with justification, which of the following series converge.
    (c) ∑ n=1 to n=∞ (n/n+1)
    You may appeal to the general limit theorems.
    Let a_{n}=\frac{n}{n+1}
    Then a_{n} \rightarrow 1 as n\rightarrow \infty
    That is, a_{n} \not\rightarrow 0 as n\rightarrow \infty and so \bigsum_{n=1}^{\infty} a_{n} is divergent.

    Sorry, I don't know the theory behind the first question but I'm sure someone else will be able to help.
    Offline

    8
    ReputationRep:
    Anyone know the tex for a right arrow with a cross through it?
    \not\to or \not\rightarrow \not\to
    Offline

    8
    ReputationRep:
    (Original post by manps)
    4.(c) Is (Z4 - { [0]4}, \odot, [1]4) a group?
    2.2 = 0 (mod 4)

    So the closure axiom isn't satisfied.
    Offline

    15
    ReputationRep:
    (Original post by Jonny W)
    \not\to or \not\rightarrow \not\to
    Thanks
 
 
 
Turn on thread page Beta
Updated: August 24, 2005

University open days

  • Sheffield Hallam University
    City Campus Postgraduate
    Wed, 17 Oct '18
  • Staffordshire University
    Nursing and Midwifery Undergraduate
    Wed, 17 Oct '18
  • Teesside University
    Undergraduate open day Undergraduate
    Wed, 17 Oct '18
Poll
If a uni gives me an unconditional offer they....
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

Equations

Best calculators for A level Maths

Tips on which model to get

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.