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Math Help - stuck in a step for finding radius and interval of convergence

  1. #1
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    stuck in a step for finding radius and interval of convergence

    find the interval and radius of convergence for sum from n=1 to infinity of \frac{x^n}{\sqrt{n}}. So I used the ratio test and I've got it down to |x| \lim_{n\to\infty}|\sqrt{\frac{n}{n+1}}|. I want to use L'Hospitals rule but I'm uncertain how to proceed. Can I use L'H rule on \frac{n}{n+1} or do I use it on \frac{\sqrt{\frac{n}{n+1}}}{1}
    Last edited by superdude; July 31st 2009 at 06:10 PM. Reason: fixed latex
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  2. #2
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    no need for L'Hopital

    \sqrt{\frac{n}{n+1}} = \sqrt{\frac{1}{1 + \frac{1}{n}}}<br />

    now take the limit as n \to \infty
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  3. #3
    Rhymes with Orange Chris L T521's Avatar
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    Quote Originally Posted by superdude View Post
    find the interval and radius of convergence for sum from n=1 to infinity of \frac{x^n}{\sqrt{n}}. So I used the ratio test and I've got it down to |x| \lim_{n\to\infty}|\sqrt{\frac{n}{n+1}}|. I want to use L'Hospitals rule but I'm uncertain how to proceed. Can I use L'H rule on \frac{n}{n+1} or do I use it on \frac{\sqrt{\frac{n}{n+1}}}{1}
    Keep in mind that \lim\left|x\right|\left|\sqrt{\frac{n}{n+1}}\right  |<1

    Now note that \sqrt{\frac{n}{n+1}}=\sqrt{\frac{1}{1+\frac{1}{n}}  }.

    So what is \lim\left|\sqrt{\frac{1}{1+\frac{1}{n}}}\right|?

    EDIT: Skeeter beat me to it...
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  4. #4
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    It's not apparent to me how
    \sqrt{\frac{n}{n+1}} = \sqrt{\frac{1}{1+\frac{1}{n}}}
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  5. #5
    No one in Particular VonNemo19's Avatar
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    Quote Originally Posted by superdude View Post
    It's not apparent to me how
    \sqrt{\frac{n}{n+1}} = \sqrt{\frac{1}{1+\frac{1}{n}}}
    Divide the numerator and denominator inside the radical on the left hand side of the equation by n and see what happens
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  6. #6
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    Quote Originally Posted by VonNemo19 View Post
    Divide the numerator and denominator inside the radical on the left hand side of the equation by n and see what happens
    does doing that sum how not effect the number? I mean you can't just go dividing things. For example, is this equivalent to multiplying by 1.
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  7. #7
    No one in Particular VonNemo19's Avatar
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    Quote Originally Posted by superdude View Post
    does doing that sum how not effect the number? I mean you can't just go dividing things. For example, is this equivalent to multiplying by 1.

    We can always multiply (in your case \frac{1}{n}) top and bottom of a fraction by the same value because it does not affect the value of the ratio that is implied.

    EG \frac{ca}{cb}=\frac{a}{b}
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  8. #8
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    got it, thanks
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