On the series expansion $\frac{x}{a+b} = \sum_{n=0}^{\infty}\frac{(-1)^nb^nx}{a^{n+1}}$

Question

Essentially, I came across the series $$\frac{x}{a+b} = \sum_{n=0}^{\infty}\frac{(-1)^nb^nx}{a^{n+1}}$$ where $|b|<|a|$, on Wolfram Alpha (https://www.wolframalpha.com/input/?i=x%2F(a%2Bb)) while looking to approximate some equations in my research. However, I have never encountered this before (and my friend in Mathematics Major has never either) so I am quite interested to know the name of this series and perhaps the proof for this expansion as well.

Thanks a lot.

Answer 1

It's just $$\frac xa\cdot\frac1{1+\frac ba}=\frac xa\sum_{k=0}^\infty \left(-\frac ba\right)^n$$ et cetera.