Is there any good reason that in educational materials, I consistently see the formula for calculating geometric series in canonical form as: $$\sum_{k=0}^{n-1} ar^k = a \frac{1-r^n}{1-r}$$
While an equally correct version is given by [edit: I have now found a book that uses this form] $$\sum_{k=0}^{n-1} ar^k =a \frac{r^n-1}{r-1}$$
The canonical form seems to be backwards for the understanding of most students, as initial examples usually have $r > 1$.
A case of special interest is the infinite series with $r<1$, with the limit of $$\sum_{k=0}^{\infty} ar^k =a\frac{1}{1-r} \text{ (when r<1)}$$
A geometric sequence is simply the discrete version of the exponential function, so a geometric series is simply the discrete version of the integral of the exponential function.
But I can't imagine that anyone would write [edit: an answer below shows that accountants do use it in this form - though I would think it is motivated by usefulness rather than consistency with the infinite sum]: $$\int_{0}^{n} ar^x dx = a\frac{1-r^n}{-\ln r} + C$$ Just because a particular definite integral is of the form: $$\int_{0}^{\infty} ar^x dx = a\frac{1}{-\ln r} \text{(when r<1)}$$
As a side note, I've never seen integration of exponentials taught with regard to geometric series, even though they are normally taught in close proximity, and the formulae are nearly identical (x-1 is a good approximation of ln x for x close to 1). Has anyone had experience with this?