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Thread: Calculus & Analysis Marathon & Questions

  1. #251
    Supreme Member seanieg89's Avatar
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    Re: First Year Uni Calculus Marathon

    Here is one instructive way to do it (probably not the fastest way):

    First choose more natural coordinates, u_n=x_n-1, v_n=y_n-2 centred at the fixed point.

    Our recurrence then transforms to:



    The matrix in question is symmetric, which implies that the eigenvalues are real and the eigenspaces are orthogonal. (An easy computation shows that there are indeed two distinct eigenvalues.)

    A consequence of this fact is that these matrix satisfies


    (Exercise).

    The maximal (maximal means maximal in absolute value btw) eigenvalue of the n-th matrix is



    and so convergence of (u_n,v_n) to 0 is proven by showing that the product of these maximal eigenvalues tends to zero, which I am sure you are capable of.



    (*) Note that this method establishes convergence for ANY choice of initial point (x_0,y_0). Note also that this is quite related to my most recently posted problem in the linear algebra marathon, which addresses sequences like this but for a fixed multiplication matrix (which is not specified and need not be symmetric).
    Last edited by seanieg89; 30 Nov 2016 at 1:28 PM.

  2. #252
    -insert title here- Paradoxica's Avatar
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    Re: First Year Uni Calculus Marathon





    Bonus: Come up with a general statement that extends to any arithmetical function that has a sufficiently "nice" asymptotic approximation.
    Last edited by Paradoxica; 6 May 2017 at 10:30 PM.
    If I am a conic section, then my e = ∞

    Just so we don't have this discussion in the future, my definition of the natural numbers includes 0.

  3. #253
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    Re: First Year Uni Calculus Marathon

    Quote Originally Posted by Paradoxica View Post




    Bonus: Come up with a general statement that extends to any arithmetical function that has a sufficiently "nice" asymptotic approximation.
    Last edited by Kingom; 6 May 2017 at 10:55 PM.

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    Supreme Member seanieg89's Avatar
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    Re: Calculus & Analysis Marathon & Questions


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