I have two formula;

$\displaystyle f(x) = \frac{a^x}{\displaystyle{\sum_{i=0}^{d-1} s^i}}$

$\displaystyle g(x) = \frac{\displaystyle{\sum_{i=1}^{x-1} a^i}}{\displaystyle{\sum_{i=0}^{x-1} a^i}}$

With 'a' being a constant, 1>a>0.

I was looking for the equivalent continuous function, that is, functions that would be exactly the same as f(x) and g(x) but would be continuous on [0, +inf

I know that these functions;

$\displaystyle f(x) = \displaystyle\frac{a^x(a-1)}{a^x-1}$

$\displaystyle g(x) = \displaystyle\frac{a^x-a}{a^x-1}$

...work fine, I'm pretty they're exactly what I'm looking for, but I have no idea how to get (with some rigor) from the discrete form to the continuous form