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Math Help - Quick Power Series Question

  1. #1
    Junior Member
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    Mar 2009
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    Quick Power Series Question

    the series is..

    [(3^n)(x+4)^n]/(n^0.5)

    did the ratio, ended up with

    3(x+4)(n^.5)/(n+1)^.5

    I got the interval of convergence correctly, but for the radius, the back of the book is saying 1/3 (when, what I'm having right now is 3x+12<1 ).

    So my question is just.. how did they get the 1/3? Do numbers not count if they're not attached to variables or did I just make a mistake
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  2. #2
    Junior Member
    Joined
    Oct 2008
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    Quote Originally Posted by coolguy99 View Post
    the series is..

    [(3^n)(x+4)^n]/(n^0.5)

    did the ratio, ended up with

    3(x+4)(n^.5)/(n+1)^.5

    I got the interval of convergence correctly, but for the radius, the back of the book is saying 1/3 (when, what I'm having right now is 3x+12<1 ).

    So my question is just.. how did they get the 1/3? Do numbers not count if they're not attached to variables or did I just make a mistake
    You did a great job. Think of radius as the maximum distance away from x. To help see this consider the following calculations.
    First of all the ratio test gives us,

    |3x+12|<1 \implies -1 < 3x + 12 < 1

    \implies -13 < 3x < -11 \implies \frac{-13}{3} < x < \frac{-11}{3}

    \implies -(4+\frac{1}{3}) < x < -(4-\frac{1}{3}) \implies

    So you can see x can move 1/3 units in either direction from -4.

    An alternate way of seeing this is from the ratio test you have

    |3(x+4)| < 1 \implies |x+4| < \frac{1}{3} (this is the way you'd do it on a test {quicker}).

    If you cannot see it by this form, then let's go one more step
    \implies -\frac{1}{3} < x+4 < \frac{1}{3} \implies -4-\frac{1}{3} < x < -4+\frac{1}{3}
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