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Math Help - [SOLVED] Power Series(Radius and interval of convergence)

  1. #1
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    [SOLVED] Power Series(Radius and interval of convergence)

    hey there..
    i've try this question already...
    i got radius of convergence, R = 1/8

    at x = 1/8,
    converges...

    at x = -1/8
    i don't how to do this one..
    please help me..
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  2. #2
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    Quote Originally Posted by nameck View Post
    hey there..
    i've try this question already...
    i got radius of convergence, R = 1/8

    at x = 1/8,
    converges...

    at x = -1/8
    i don't how to do this one..
    please help me..

    At x=-\frac{1}{8} you get \sum\limits_{n=1}^\infty\frac{(-1)^n}{n!} = a Leibnitz series and thus convergent.

    Tonio
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  3. #3
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    the question is..
    find the radius of convergence and interval of convergence of the
    power series (attachment)

    tonio.. is (8^n) / (-8) = -1^n?
    is my radius of interval is correct?
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  4. #4
    No one in Particular VonNemo19's Avatar
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    a_n=\frac{8^n}{n!}x^n

    |\frac{a_{n+1}}{a_n}|=|\frac{8^{n+1}}{(n+1)!}x^{n+  1}\cdot\frac{n!}{8^nx^n}|=\frac{8|x|}{n+1}

    Now, \lim_{n\to\infty}\frac{8|x|}{n+1}=0<1 \forall{x}\in(-\infty,\infty).

    Therefore, the series converges \forall{x} by the ratio test.

    This implies an infinte radius.
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  5. #5
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    how did u get radius from infity to -infinity?
    i haven't learn this kind solution(ration test to find radius and interval)
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  6. #6
    No one in Particular VonNemo19's Avatar
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    Quote Originally Posted by nameck View Post
    how did u get radius from infity to -infinity?
    i haven't learn this kind solution(ration test to find radius and interval)
    Well, what tests for convergence do you know?
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  7. #7
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    Quote Originally Posted by nameck View Post
    the question is..
    find the radius of convergence and interval of convergence of the
    power series (attachment)

    tonio.. is (8^n) / (-8) = -1^n?

    No, and this is not what you have. At x=-\frac{1}{8} you have \frac{8^n}{n!}\left(-\frac{1}{8}\right)^n=\left(\frac{8}{-8}\right)^n\frac{1}{n!}=\frac{(-1)^n}{n!}

    is my radius of interval is correct?
    No. The convergence radius is what VonNemo wrote and for the reasons he/she wrote.

    Tonio
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