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Homework Help: Convergence questions

  1. Nov 9, 2007 #1


    this pictures have a high quality ,you can zoom it.

    in this links i show the question and how i tried to solve them.
    i was taught this convergence stuff using this type questions.
    i solve them like that
    a = (a +6)^0.5
    n+1 n

    a1=(6)^0.5 n=>1 are given

    find if the goes up or down
    and if its converges and to what limit??

    i know that An+1 and An has the same limit
    so i put L (limit) instead of them
    and get that there are two possible limits
    L=1 L=4

    know i want to proove that this series goes up:
    i fing A2 using A1=(6)^0.5

    than using an inequality An+1>An (we assume that it is correct)
    i prooved An+2>An+1

    and prooved that the function goes up

    than the same way i prooved than An<4 (converges to 4).

    but in this questions that i added in the link
    they dont follow this pattern
    i tried to solve them that way
    but they are different
    how do i solve them??
  2. jcsd
  3. Nov 9, 2007 #2


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    The behavior of the first sequence is closely related to the behavior of the function f(x)=(x^2+2)/3, since a_(n+1)=f(a_n). Eg the sequence will be increasing when a_n is in a region where f(x)-x is positive. Try thinking along those lines. Similarly for 4/(x-5). For the other two remember if you are applying some like the ratio test, the ratio doesn't have to be <1 for ALL n. Just for LARGE n.
    Last edited: Nov 9, 2007
  4. Nov 9, 2007 #3


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    In the first problem (the imageshack problem) you are concerned with the convergence of the series (i.e. sum), are you not? You have correctly arrived at the ratio
    [tex]\frac{a_{n+1}}{a_n}= \frac{n+10}{2n+1}[/tex]
    What is important is the limit of that as n goes to infinity: you can see that easily if you divide both numerator and denominator by n:
    [tex]\frac{n+10}{2n+1}= \frac{1+\frac{10}{n}}{2+ \frac{1}{n}}[/tex]
    As n goes to infinity, those fractions with n in the denominator go to 0 and you can easily determine whether the entire limit is less than or greater than 1.
  5. Nov 9, 2007 #4
    i know that i can make a limit problem from it
    but the questions type ask for (as i wrote in the example)

    to proove using an inequality if the function is going up or down
    and then proove using a second inequality to what limit its going to.

    i showed in the example a type of problem
    but how i proove in the same way
    the other types of questions?
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