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Math Help - My calc book is very vague

  1. #1
    MHF Contributor Mathstud28's Avatar
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    My calc book is very vague

    How do you find the convergence radius of a taylor polynomial

    such as \sum_{n=0}^infty\frac{n^2(x^(2n))}{3^(n-1)}
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  2. #2
    MHF Contributor Mathstud28's Avatar
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    If this doesn't work I give up

    \sum_{n=0}^{\infty}\frac{n^2x^{2n}}{3^{n-1}}\
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  3. #3
    Behold, the power of SARDINES!
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    Quote Originally Posted by Mathstud28 View Post
    \sum_{n=0}^{\infty}\frac{n^2x^{2n}}{3^{n-1}}\

    Try the ratio test.

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

    if L < 0 the series converges
    if L=1 the test is inconclusive
    if  L\ge 1 The series diverges

    \lim_{n \to \infty}\left| \frac{\frac{(n+1)^2x^{2(n+1)}}{3^{(n+1)-1}}}{\frac{n^2x^{2n}}{3^{n-1}}}\right|

    \lim_{n \to \infty}\left| \frac{(n+1)^2x^{2n+2}}{3^{n}} \cdot \frac{3^{n-1}}{n^2x^{2n}}\right|

    \lim_{n \to \infty}\left| \frac{(n+1)^2x^2}{3n^2}\right|=\left|\frac{x^2}{3} \right|

    So we want the series to converge so we want

    \left| \frac{x^2}{3}\right| < 1

    if we solve the inequality we get the radius of convergance.

    Note you need to check the end points with another test.
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  4. #4
    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by Mathstud28 View Post
    \sum_{n=0}^{\infty}\frac{n^2x^{2n}}{3^{n-1}}\
    Quote Originally Posted by Mathstud28 View Post
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  5. #5
    MHF Contributor Mathstud28's Avatar
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    So

    Obviously we want a_{n+1}\< a_{n}\ because then that is what causes convergence?
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  6. #6
    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by Mathstud28 View Post
    Obviously we want a_{n+1}\< a_{n}\ because then that is what causes convergence?
    no, we want \lim_{n \to \infty} \left| \frac {a_{n + 1}}{a_n} \right| < 1 ............then we have convergence by the ratio test

    all you need to do is solve the last line TheEmptySet left you with for x. be sure to check the end-points as well
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  7. #7
    MHF Contributor Mathstud28's Avatar
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    So

    if its not too much to ask could you give me an example I work through it and you see if I am right? If thats too much work I understand...with all that LaTeX you have to go through to do it I wouldn't blame you haha
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