About series representations

Question

Can $\displaystyle \frac{1}{1+\frac{z}{n}}$ have the following series representation?

$\displaystyle \sum_{k=0}^{\infty}\frac{(-z)^k}{n^k}$

Answer 1

If $\left|\frac{z}{n}\right|\lt 1$, yours is indeed the correct series respresentation. We have here a particular example of a geometric series.

Answer 2

In general for any $w\in \mathbb C$ with $|w|<1$ it holds that $\frac{1}{1-w}=\sum_{k=0}^\infty w^k$ (this is the sum of a geometric series). Thus, for $w=-\frac{z}{n}$, if $|w|<1$, then plugging that $w$ into the equation above yields the affirmative answer to your question.