I have a summation $\sum\limits_{k = -\infty}^{\infty} r^k$ that I've split up to $\sum\limits_{k = -\infty}^{-1} r^k + \sum\limits_{k = 0}^{\infty} r^k$. The second summation is a solvable geometric series, which gives me $\frac{1}{1 - r}$.
To compute the second summation, I am trying to change the variable index and use the same geometric series. I set $a = -k$ and use $\sum\limits_{a = 0}^{\infty} r^a$. This over-counts by one, but subtracting out the zero term at the end is easy.
My question is, how does this change of variable affect the value with respect to $k$? Surely this half of the summation is not also equal to $\frac{1}{1 - r}$ (- 1 to account for the extra 0 term). What would its value be instead?