How to explain the formula for the sum of a geometric series without calculus?

sequences-and-series summation education learning

This could be explained using algebraic transformation but i would rather show a very simple geometric proof for sum:

1 + 1/2 + 1/4 + ... = 2

enter image description here


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.

Related

Continuous extension of a uniformly continuous function from a dense subset.
Modular Arithmetic - Find the Square Root
How to prove that $\lim\limits_{n \to \infty} \frac{k^n}{n!} = 0$ [duplicate]
Are derivatives defined at boundaries?
The absolute value of a Riemann integrable function is Riemann integrable.
How to calculate: $\sum_{n=1}^{\infty} n a^n$ [duplicate]
FoxTrot Bill Amend Problems
Showing that $\int_0^\infty x^{-x} \mathrm{d}x \leq 2$.
Why does $a^n - b^n$ never divide $a^n + b^n$?
On orders of a quadratic number field
If $p$ is a polynomial, then either $\,p(z)=z^n\,$ or $\,\max_{|z|=1}|p(z)|>1$.
Pointwise a.e. convergence and weak convergence in Lp