Workshop: Session 6 — Summary

We began by looking at some of the lovely mathematics of infinite series. Working with the geometric series

$\frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \frac{1}{16} + \ldots$

we explored partial sums, the formula for the sum of an infinite geometric series, and some ideas in Calculus.

By looking at the partial sums of this series, a lot of good ideas surface:

$\frac{1}{2}=\frac{1}{2}$

$\frac{1}{2} + \frac{1}{4}=\frac{3}{4}$

$\frac{1}{2} + \frac{1}{4} + \frac{1}{8}=\frac{7}{8}$

You can quickly generalize to the formula

$\frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \frac{1}{16} + \ldots + \frac{1}{2^n} = \frac{2^n-1}{2^n}$

With this, you can predict the answer (1!) and discuss ideas like limits.

Still working with the above series, you can explore the way in which we derive the formula for the sum of an infinite gepmetric series.

Take the original series, set it equal to S

$\qquad\quad \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \frac{1}{16} + \ldots = S$ (Eq 1)

multiply this equation by 2

$2( \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \frac{1}{16} + \ldots = S)$

to get

$1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \frac{1}{16} + \ldots = 2S$ (Eq 2)

Then subtract Eq1 from Eq2 to get the magical result

$S = 1$

A nice geometric interpretation of this can be seen in this Proof Without Words.

The same approach produces the general formula for the sum of infinite geometric series:

$a + ar + ar^2 + ar^3 + ar^4 + \ldots = \frac{a}{1-r}$

We also had some fun looking at series where r, the common ratio, didn’t satisfy the requirement $|r| < 1$ .