I was thinking about Supertasks (see the Wikipedia article), and wanted to be able to find the total amount of time the Supertask takes using an equation.
This is what I came up with:
$$a + \sum_{n=1}^{\infty}\left(\frac{x}{y}\right)^na = \frac{y}{y-x}a$$
Is this true? If not, is it true when $x = 1$ and $y = 2$?
When $|x|<|y|$, then $|x/y|<1$ and we can use the usual geometric series formula $$\sum_{n=0}^\infty r^n = \frac1{1-r}, \quad \text{provided } |r|<1.$$ This means that $$\sum_{n=1}^\infty r^n = \frac r{1-r}.$$ Substitute $r=x/y$ and you almost get your formula, but there's a slight error. Note that $$\frac r{1-r} = \frac{\frac xy}{1-\frac xy} = \frac x{y-x}.$$
When $|x|\ge |y|$, the sum does not converge.