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.
Now, let’s see a similar 3-dimensional proof that 1/8 + 3/16 + 6/32 + 10/64 + 15/128 + … = 1
I will leave that as an exercise for the reader.
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
I love it!
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.