I'm staring at
$$ f(n) = \sum_{x=1}^n a(n)^{n-x}\\ a(n) = \frac{(1 - \frac{f}{n})y}{y(1 - \frac{f}{n} - \frac{\delta}{n}) + \frac{\delta}{n}}$$
where $f$, $y$, $\delta$, all $ \in (0, 1)$, and $n$ is a positive integer.
and want to let $\lim_{n\to\infty} f(n)$. Wolfram alpha told me that
$$ \sum_{x=1}^n b^{n-x} = \frac{b (b^n - 1)}{b - 1}$$
for any constant $b$ - and since $x$ does not appear in the sum I assume I can apply that here as well. But I'm not sure how to proceed now. It looks very similar to one of the definitions of the exponential - but I could't come up with a way to use that.