## Thursday, 10 February 2011

### Infinite Series and Proofs w/o Words

Came across a nice post at irrational cube about proofs- without words and infinite series....

here is a teaser, and a link to the  post.

Sometimes I inadvertently come up with a math problem and can’t seem fall asleep until I solve it.  So it happened last night.
It started with my unit on limits.  I plan on giving the students a math problem in which they fill in half of a square, half of the remaining area, and so on and so forth until they see that they get arbitrarily close to filling in the whole thing.  Mathematically, this means that
$\displaystyle \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \frac{1}{16} + ... = 1$
or
$\displaystyle \sum_{n=1}^{\infty} \frac{1}{2^n} = 1$
I made it into a proof without words for the visually inclined:
(note: the different color shading is to differentiate iterations)

and finishing up with .......
$\displaystyle \frac{1}{x} + \frac{1}{x^2} + \frac{1}{x^3} + \frac{1}{x^4} + ... = \frac{1}{x-1}$

Really nice... so check it out﻿