Question about infinity sum

In a problem about calculating the Laurent series of an expresion, I'm seeing:

$$\frac{1}{1+\frac{1}{w}}=\sum_{k=0}^\infty \frac{(-1)^k}{w^k}$$ When does that hold? What are the conditions of $1/w$ for that to be true? I guess that in a more general way: $$\sum_{k=0}^\infty (-1)^kz^k=\frac{1}{1+z}$$ Again, what are the conditions on $z$?

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