Is there a way to simplify the following expression?
$${\sum_{i=1}^n (1-g(x))^{i-1}} \over {\sum_{i=1}^n (1-g(x))^i} $$
With $0<g(x)<1$ and $n \in \mathbb{N} $. This expression is the same as:
$${\sum_{i=0}^{n-1} (1-g(x))^{i}} \over {\sum_{i=1}^n (1-g(x))^i} $$
Let's say $(1-g(x))=h(x)$. I can rewrite the above expression as:
$${1-h(x)^n\over {1-h(x)}}\over {{h(x)^{n+1}-h(x)} \over {1-h(x)}} $$
$${1-h(x)^n}\over {h(x)^{n+1}-h(x)} $$
$${1-h(x)^n}\over {h(x)(h(x)^n-1) } $$
But this does not really help me to make it any clearer for me.
Is there another simplification I'm missing?
Best Answer
Found a small mistake, note that $$\sum_{k=1}^n x^k = \frac{x^{n+1}-x}{x-1}$$ and so we have a sign error. To fix this, just tack on a negative sign. This leads to:
$$\frac{h(x)^n-1}{h(x)^{n+1}-h(x)} = \frac{1}{h(x)} \cdot \frac{h(x)^n-1}{h(x)^n-1} = \frac{1}{h(x)} = \frac{1}{1-g(x)}$$