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How do I obtain the Taylor expansion of $$\frac{1}{1+z}$$ about $a=i$ please? Do I just expand the series using the binomial expansion?

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Here is a start:

$$ \frac{1}{1+z} = \frac{1}{(1+i)+(z-i)}= \frac{1}{1+i} \frac{1}{1+\frac{(z-i)}{1+i}} = \dots\,.$$

You need to use the geometric series to expand the last expression.

Note:

$$ \frac{1}{1+t} = \sum_{k=0}^{\infty} (-1)^k t^k, \quad |t|<1.$$

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  • $\begingroup$ @Timbuc: You are right! But as you see I gave him these things in the note. Thanks for your comment. $\endgroup$ – Mhenni Benghorbal Jan 21 '15 at 10:18
  • $\begingroup$ how did you obtain the third expression please? $\endgroup$ – user120768 Jan 21 '15 at 21:57
  • $\begingroup$ I just factored out $1+i$. $\endgroup$ – Mhenni Benghorbal Jan 22 '15 at 1:46

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