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Math Help - radius and interval of convergence

  1. #1
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    radius and interval of convergence

    could someone help me with this problem? stuck on homework added the print screen of the problem
    radius and interval of convergence-problem.jpg
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  2. #2
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    Quote Originally Posted by pham07 View Post
    could someone help me with this problem? stuck on homework added the print screen of the problem
    Click image for larger version. 

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    a_n=\frac{n(x-2)^n}{n^2+1}\Longrightarrow\left|\frac{a_{n+1}}{a_  n}\right|=\left|x-2\right|\frac{n}{n+1}\frac{(n+1)^2+1}{n^2+1} \xrightarrow [n\to\infty]{}|x-2|<1\Longleftrightarrow 1<x<3

    From here you get the radius of convergence and almost the interval of convergence: you just have to check whether the extreme points 1, 3 belong to it or not.
    Now you do the second one by yourself.

    Tonio
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  3. #3
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    alright thanks.. I did it by myself and this is what I got.. did I do it correctly?
    radius and interval of convergence-problem2.jpg
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  4. #4
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    Quote Originally Posted by pham07 View Post
    alright thanks.. I did it by myself and this is what I got.. did I do it correctly?
    Click image for larger version. 

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    Well...almost, if I understood what you meant with those weird arrows:

    \left|\frac{a_{n+1}}{a_n}\right|\xrightarrow[n\to\infty]{}\frac{1}{4}|x+1|<1\Longleftrightarrow |x+1|<4\Longleftrightarrow -5<x<3 and etc.

    Tonio
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