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Math Help - radius of convergence problem

  1. #1
    Junior Member
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    radius of convergence problem

    n!(x+3)^n
    -----------
    4^n

    I do all the work, and come up with some ridiculous fraction:

    (n+1)(1/16)^n*|x+3|
    ___________________
    4

    Take limit as n-> inf and I get zero?

    I used maple as well to check and it gets:

    (n+1)!(x+3)^(n+1)*4^n
    _____________________
    4^(n+1)*n!(x+3)

    = 1/4|(x+3)(n+1)|
    take limit and it becomes infinite.

    Which is right?
    note:
    I did not show all my steps. i used the ratio test on the original problem and as you guys know, replace all n with n+1, times by reciprocal of fraction etc, You then get what i have above. I used a calculator for mine, and then tried to duplicate it with maple but get two different answers. What I need to do is find the radius of convergence.
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  2. #2
    MHF Contributor alexmahone's Avatar
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    Maple is right. The radius of convergence is 0.
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  3. #3
    MHF Contributor
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    Quote Originally Posted by orendacl View Post
    n!(x+3)^n
    -----------
    4^n

    I do all the work, and come up with some ridiculous fraction:

    (n+1)(1/16)^n*|x+3|
    ___________________
    4

    Take limit as n-> inf and I get zero?

    I used maple as well to check and it gets:

    (n+1)!(x+3)^(n+1)*4^n
    _____________________
    4^(n+1)*n!(x+3)

    = 1/4|(x+3)(n+1)|
    take limit and it becomes infinite.

    Which is right?
    note:
    I did not show all my steps. i used the ratio test on the original problem and as you guys know, replace all n with n+1, times by reciprocal of fraction etc, You then get what i have above. I used a calculator for mine, and then tried to duplicate it with maple but get two different answers. What I need to do is find the radius of convergence.
    You need to find the values of x for which

    \lim_{n \to \infty}\left|\frac{a_{n + 1}}{a_n}\right| < 1

    \lim_{n \to \infty}\left|\frac{\frac{(n + 1)!(x + 3)^{n + 1}}{4^{n + 1}}}{\frac{n!(x + 3)^n}{4^n}}\right| < 1

    \lim_{n \to \infty}\left|\frac{4^n(n + 1)!(x + 3)^{n + 1}}{4^{n + 1}n!(x + 3)^n}\right| < 1

    \lim_{n \to \infty}\left|\frac{(n + 1)(x + 3)}{4}\right| < 1

    \left|\frac{x + 3}{4}\right|\lim_{n \to \infty}(n + 1) < 1.

    \left|\frac{x + 3}{4}\right| < \frac{1}{\lim_{n \to \infty}(n + 1)}

    \left|\frac{x + 3}{4}\right| < 0.


    It is impossible to have a size less than 0, so there is not anywhere where the series converges. In other words, the radius of convergence is 0.
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