Proof Without Words: Two Dimensional Geometric Series

I offer this visual proof of the following amazing infinite sum

$\frac{1}{4} + \frac{2}{8} + \frac{3}{16} + \frac{4}{32} + \frac{5}{64} + ... = 1$

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Graeme McRae · April 30, 2013 at 10:25 am

Now, let’s see a similar 3-dimensional proof that 1/8 + 3/16 + 6/32 + 10/64 + 15/128 + … = 1

MrHonner · April 30, 2013 at 11:24 am

I will leave that as an exercise for the reader.

Mike Lawler · May 2, 2013 at 6:42 pm

Here’s two attempts at the 3d visual proof that 1/8 + 3/16 + 6/32 + 10/64 + . . . = 1 using Lego Digital Design. It was neat to see the cube coming together while building this. The first video is the first three terms, and the 2nd is the first 5.

I love it!

Nat Silver · May 10, 2013 at 8:30 pm

JIm Loy sums the series on his website, using partitions:
http://www.jimloy.com/algebra/aseries.htm A standard approach would be
S = (1/4)(1+2x+3x^2+4x^3+…) where x = 1/2.
S = (1/4)(d/dx(x+x^2+x^3+…))
= (1/4)(d/dx(x/(1-x)))
= (1/4)(1/(1-x)^2) where x = 1/2,
as I’m sure you are aware.

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