## Exploding Dots

### 1.9 An Infinite Process: The Geometric Series Formula

Lesson materials located below the video overview.

Consider again an $$1 \leftarrow x$$ base machine.

We can use this machine to divide $$1$$ by $$1-x$$, that is, to compute $$\dfrac{1}{1-x}$$.

The quantity “$$1$$” is a single dot in the units position and the quantity “$$1-x$$” is an anti-dot in the $$x$$ position.

We wish to find copies of   in the picture . Of course there are none at this stage.

The trick is to fill and empty box with a dot and anti-dot pair. This gives us a copy of   in the units position.

We can repeat this trick:

and again, infinitely often!

This shows that, as a statement of algebra, we have:

### $$\dfrac{1}{1-x}=1 + x + x^{2} + x^{3} + …$$

 Question 45:   a)    Use this technique to show that $$\dfrac{1}{1+x} = 1 – x + x^{2} – x^{3} – x^{4} + …$$.     b)    Compute $$\dfrac{x}{1-x^{2}}$$.     c)     Compute $$\dfrac{1}{1 – x – x^{2}}$$ and discover the Fibonacci numbers!

Even more fun thoughts about the geometric series appears in the COMPANION GUIDE to this EXPLODING DOTS course – as well as some commentary as to when the formula is true as a statement of arithmetic, not just a statement of algebra.

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