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
\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

1 comment:

Steven Colyer said...

Very nice.

He's another example about algebraic geometry, and something very fresh, that also involves infinite series and polygons, in this case how many triangles exist in square tilings cut crosswise into triangles:

Counting Triangles on a tin ceiling by Dave Richeson at his "Divison by Zero" blog.

It's also important in that it points out the important interplay between teacher and student.

See if you can figure that out, and if not, Dave gives the answer in his next blogpost: here.

This is actually new and therefore on-going research, which means someone reading this can either further solve (or expand on) the solution, or find if this has been done before.

Which is kinda neat, either way.