Volume Measure

Strangely, the Menger Sponge has a volume measure of zero. That is, the volumes of the parts eliminated in the construction of the sponge sum to as much volume as was originally there. In the first step, we removed 7/27 of the volume. In the next step, we removed 7 sub-subcubes each having volume 1/272 from each of 20 subcubes for a total of 7(20/272) removed. Continuing in this way, we find that the total fraction of the volume removed was

7/27 + 7(20/272) + 7(202/273) + ... + 7(20(n-1)/27n) + ...

The following cell produces a table of partial sums s[n] of this series for n= 5,10,...60.

Input := 

Clear[n,a,k]
a[k_]:=7(20^(k-1)/27^k)
s[n_]:=Sum[a[k],{k,1,n}]//N;
Table[{n,s[n]},{n,5,60,5}]//TableForm

Output =

5    0.776986

10   0.950265

15   0.988908

20   0.997526

25   0.999448

30   0.999877

35   0.999973

40   0.999994

45   0.999999

50   1.

55   1.

60   1.

Clearly, this series converges to 1, so the sponge is left with zero volume. The sum representing the total volume removed could also be calculated analytically using the fact that

1/(1-x) = 1 + x + x2 + ..., if |x|<1.