Series convergence

**splash** · November 2nd 2006, 07:49 PM

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.

**ThePerfectHacker** · November 2nd 2006, 07:58 PM

Originally Posted by splash

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

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