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?

As I understand your question, your student know and understand the formula for a finite geometric sum.
You just need to convince them that the sum goes to $\frac 1{1-q}$  as $n \to \infty$.
Well the only thing left to do really is to convince them that if $x < 1$, then $x^n \to 0$.
You can do so by asking them to bring their calculator to the class, hit $1/2 \cdot 1/2$.
Then again $1/2$. Then again $1/2$. After 10 or 20 iterations you can write this number out with decimal digits and they should grasp the fact that it's very close to $0$ (better to write the number out than to be left with something like $\frac 1{2^{16}}$ which may be less clear) 
