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Math Help - Series convergence

  1. #1
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    Series convergence

    Show that \sum_{n=0}^{\infty} \frac{(z+i)^n}{2^n} converges for all values of z in the disk D(-1)={z:abs(z+i)<2} and diverges if abs(z+i)>2.

    abs() means absolute value.
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  2. #2
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    Quote Originally Posted by splash View Post
    Show that \sum_{n=0}^{\infty} \frac{(z+i)^n}{2^n} converges for all values of z in the disk D(-1)={z:abs(z+i)<2} and diverges if abs(z+i)>2.

    abs() means absolute value.
    This is a power series centered at -i
    Use the generalized ratio test for power series.
    That is,
    \lim_{n\to\infty}\frac{(1/2)^{n+1}}{(1/2)^n}=1/2
    That means the series converges absolutely when |z+i| is strictly less then reciprocal of 1/2=2.

    Thus,
    |z+i|<2---> converges (absolutely)

    |z+i|>2---> diverges
    Last edited by ThePerfectHacker; November 3rd 2006 at 07:33 AM.
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