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Math Help - recursive sequences!

  1. #1
    Member javax's Avatar
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    Post recursive sequences!

    hey guys...I'm having a big trouble with Recursive Sequences(Sequences and Series in Majority)! I just failed on my first exam because of not having a clear idea about them(their convergence, sum..etc) I was asking if you could send me any link or suggest me a book from where I can get better understanding!

    thnx for your time
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    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by javax View Post
    hey guys...I'm having a big trouble with Recursive Sequences(Sequences and Series in Majority)! I just failed on my first exam because of not having a clear idea about them(their convergence, sum..etc) I was asking if you could send me any link or suggest me a book from where I can get better understanding!

    thnx for your time
    what about your text? did you try google or wikipedia?

    remember, you can also ask specific questions here and we will help you
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    Member javax's Avatar
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    Quote Originally Posted by Jhevon View Post
    what about your text? did you try google or wikipedia?

    remember, you can also ask specific questions here and we will help you
    Yeah I tried google but I'm not finding the 'core'.
    The problem is that my professor didn't make much excersises on this particular issue! On the other hand there are such a probs. on tests!
    for e.x. like this one:


    or another example:


    too messy
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    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by javax View Post
    Yeah I tried google but I'm not finding the 'core'.
    The problem is that my professor didn't make much excersises on this particular issue! On the other hand there are such a probs. on tests!
    for e.x. like this one:
    show that the sequence is monotonically decreasing and that it is bounded below by 0. this will show it converges. you can do this using induction

    for the second part, call the limit L and note that \lim x_n = \lim x_{n + 1}

    so, \lim x_{n + 1} = \lim \frac {\sqrt{x_n + 1} - 1}{x_n} \implies L = \frac {\sqrt{L^2 + 1} - 1}L

    now solve for L
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    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by javax View Post
    or another example:


    too messy
    again, assuming the limit exists, call it L

    taking the limit of the system y_n = y_{n - 1}(2 - xy_{n - 1}) we get:

    L = L(2 - xL)

    now solve for L. disregard one of the solutions based on a reasonable cause
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    Member javax's Avatar
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    Quote Originally Posted by Jhevon View Post
    again, assuming the limit exists, call it L

    taking the limit of the system y_n = y_{n - 1}(2 - xy_{n - 1}) we get:

    L = L(2 - xL)

    now solve for L. disregard one of the solutions based on a reasonable cause
    you gotta help me with the L a bit more to finalize!
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    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by javax View Post
    you gotta help me with the L a bit more to finalize!
    help in what way? explain your set back. you mean to solve for L, or are you just not getting the whole L thing altogether?
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  8. #8
    Member javax's Avatar
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    Quote Originally Posted by Jhevon View Post
    help in what way? explain your set back. you mean to solve for L, or are you just not getting the whole L thing altogether?
    as I understand you're supposing that there exists the limit of Xn, named(L)
    if it is for Xn, it means it is for Xn-1 too!
    But then?
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    Quote Originally Posted by javax View Post
    as I understand you're supposing that there exists the limit of Xn, named(L)
    The first step in this is to prove that limits do exist!
    You have not done that.
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    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by Plato View Post
    The first step in this is to prove that limits do exist!
    You have not done that.
    indeed. as i said javax, induction is the way i usually see this done. and you are to show the sequence is monotonic and bounded, because then, a theorem tells us it converges (by the way, it is in proving the limit exist that you realize which of the answers in the second part to disregard)

    Quote Originally Posted by javax View Post
    as I understand you're supposing that there exists the limit of Xn, named(L)
    if it is for Xn, it means it is for Xn-1 too!
    But then?
    yes. so when we take the limit, all the y_n's go to L, and we get the expression i have there

    now we must solve for L. begin by expanding the right hand side, and then try to get all the L's on one side to solve for L
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  11. #11
    Member javax's Avatar
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    Quote Originally Posted by Plato View Post
    The first step in this is to prove that limits do exist!
    You have not done that.
    assume & then prove(by induction or smth)! I think it goes like that but I better consult with books now...
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    Quote Originally Posted by Jhevon View Post
    indeed. as i said javax, induction is the way i usually see this done. and you are to show the sequence is monotonic and bounded, because then, a theorem tells us it converges (by the way, it is in proving the limit exist that you realize which of the answers in the second part to disregard)

    yes. so when we take the limit, all the y_n's go to L, and we get the expression i have there

    now we must solve for L. begin by expanding the right hand side, and then try to get all the L's on one side to solve for L
    Thanks, I think I got your point...I'll see a bit more about induction
    Thanks again!
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    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by javax View Post
    Thanks, I think I got your point...I'll see a bit more about induction
    Thanks again!
    sure. if you get stuck, come back

    what's smth?
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  14. #14
    Member javax's Avatar
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    ahhh, I meant 'something'
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