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Math Help - Calc BC(2?) Series Question

  1. #1
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    Calc BC(2?) Series Question

    My high school teacher gave us the following problem:

    The coefficients of the power series (starting at n=0 and continuing for all real numbers) an(x-2)n satisfy a0 = 5 and an = ((2n+1)/(3n+1))an-1 for all n≥1. What is the radius of convergence of the series?

    I've been taught to use the "ratio" test. Effectively placing the n+1 term over the nth term, simplifying, and then evaluating. Based on my work I believe that the answer should be 0, but am unsure whether it is correct due to that fact that I was absent when the material was taught.

    P.S. Does anyone know how to write sigma notation in a word processor or in html?
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  2. #2
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    Re: Calc BC(2?) Series Question

    Quote Originally Posted by Xeritas View Post
    My high school teacher gave us the following problem:

    The coefficients of the power series (starting at n=0 and continuing for all real numbers) an(x-2)n satisfy a0 = 5 and an = ((2n+1)/(3n+1))an-1 for all n≥1. What is the radius of convergence of the series?

    I've been taught to use the "ratio" test. Effectively placing the n+1 term over the nth term, simplifying, and then evaluating. Based on my work I believe that the answer should be 0, but am unsure whether it is correct due to that fact that I was absent when the material was taught.

    P.S. Does anyone know how to write sigma notation in a word processor or in html?
    note that the ratio test can also be done using the ratio of the nth term to the (n-1)st term ...

    \lim_{n \to \infty} \left|\frac{a_n(x-2)^n}{a_{n-1}(x-2)^{n-1}}\right| < 1

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

    |x-2| \lim_{n \to \infty} \frac{2n+1}{3n+1} < 1

    |x-2| \cdot \frac{2}{3} < 1

    |x-2| < \frac{3}{2}
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    Re: Calc BC(2?) Series Question

    Thanks, I didn't think of that.
    Last edited by Xeritas; March 15th 2012 at 06:52 PM. Reason: To not have to post again...
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