How to explain the formula for the sum of a geometric series without calculus? How to explain to a middle-school student  the notion of a geometric series without any calculus (i.e. limits)? For example I want to convince my student that
$$1 + \frac{1}{4} + \frac{1}{4^2} + \ldots + \frac{1}{4^n} = \frac{1 - (\frac{1}{4})^{n+1} }{ 1 - \frac{1}{4}}$$
at $n \to \infty$ gives 4/3?

What about multiplying the LHS by $(1 - \frac{1}{4})$?
Or is that what you wanted to avoid?
I mean it is not so difficult to understand that nearly all terms cancel... ?
