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.

ex1901

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.

ex1902

 

We wish to find copies of ex1903  in the picture ex1904. 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  ex1903 in the units position.

ex1905

We can repeat this trick:

ex1906

and again, infinitely often!

ex1907

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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