Can anyone help with this question?

Find the power series expansions of

(a) log z about z = 1;

(b) 1/(1 + z) about z = −5.

What is the radius of convergence in each case?

Thanks for any help

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- Feb 10th 2010, 01:12 AMalawrieComplex Power Series
Can anyone help with this question?

Find the power series expansions of

(a) log z about z = 1;

(b) 1/(1 + z) about z = −5.

What is the radius of convergence in each case?

Thanks for any help - Feb 10th 2010, 03:27 AMmr fantastic
For (a), where are you stuck? What have you tried?

(b) Note that $\displaystyle \frac{1}{1 + z} = \frac{1}{(z + 5) -4}$.

**Region I:**$\displaystyle 0 < \frac{|z + 5|}{4} < 1$.

$\displaystyle \frac{1}{(z + 5) -4} = \frac{-1}{4 - (z + 5)} = -\frac{1}{4} \left( \frac{1}{1 - \left(\frac{z+5}{4}\right)} \right)$ and you can get a series using the sum of an infinite geometric series, noting that $\displaystyle r = \frac{z + 5}{4}$.

**Region II:**$\displaystyle \frac{|z + 5|}{4} >1$.

$\displaystyle \frac{1}{(z + 5) -4} = \frac{1}{z + 5} \left( \frac{1}{1 - \left[ \frac{4}{z + 5}\right]} \right)$ and you can get a series using the sum of an infinite geometric series, noting that $\displaystyle r = \frac{4}{z + 5}$.