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?

I think a 14 year old can grasp the fact that $$\frac 12 + \frac 14 + \frac 18 + \frac 1{16} + \cdots = 1$$ rather intuitively. (Go halfway there, then half the remaining distance, then halfway again, and so on and you get arbitrarily close....)
If you are willing to do a little algebra (and wave hands about rearrangement) you get
$$
2 \left( \frac 14 + \frac 1{16} + \cdots  \right) + \left( \frac 14 + \frac 1{16} + \cdots \right) = 1$$
so that
$$
\frac 1{4} + \frac{1}{16} + \cdots = \frac 13.
$$
Now add 1.
