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.

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.

I wrote a column for Quanta Magazine on the recently discovered “hat tile”, the first ever aperiodic monotile! Have you ever admired how the slats of a hardwood floor fit together so cleanly, or how Read more…

In my latest column for Quanta Magazine I combine my love of geometric dissections with my appreciation of The Great British Bake Off. Gina the geometry student stayed up too late last night doing her Read more…

## 5 Comments

## 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.

https://www.youtube.com/watch?v=F0nvNtuKQBA

## Graeme McRae · May 2, 2013 at 7:04 pm

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.