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Math Help - Intervals of convergence of power series

  1. #1
    bsteel
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    Intervals of convergence of power series

    I need some homework help for finding the interval of convergence for the series (x/2)^n where n starts at n=0 and ends at n=infinity.
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by bsteel
    I need some homework help for finding the interval of convergence for the series (x/2)^n where n starts at n=0 and ends at n=infinity.
    The series you are interested in the convergence of may be written:

    <br />
\sum_{n=0}^{\infty} (x/2)^n=\lim_{N\rightarrow \infty}\sum_{n=0}^{N} (x/2)^n<br />

    Now:

    <br />
\left(\sum_{n=0}^{N} (x/2)^n \right)(1-(x/2))=1-(x/2)^{N+1}<br />

    (If you multiply out the product on the Left Hand Side you will find
    all but the first and last terms cancel).

    So:

    <br />
\sum_{n=0}^{N} (x/2)^n=\frac{1-(x/2)^{N+1}}{1-(x/2)}<br />

    Hence

    <br />
\lim_{N\rightarrow \infty}\sum_{n=0}^{N} (x/2)^n<br />

    converges if and only if:

    <br />
\lim_{N\rightarrow \infty} \frac{1-(x/2)^{N+1}}{1-(x/2)}<br />

    converges, and this last expression converges if and only if

    <br />
\left| \frac{x}{2}\right| <1<br />

    or, if and only if:

    <br />
-2<x<2<br />

    RonL
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