High h/k per serving,
Part n radius: n*r/k
Bottom area of part n: pi * n 2 * r 2/k 2
Volume n: pi * h * n 2 * r 2/k 3
Total volume (1+2+3+4+5+...+n): pi * h * (12+3 2+4 2+...+k 2) * r 2/k 3.
because
1^2+2^2+3^2+4^2+...+k^2=k*(k+ 1)*(2k+ 1)/6
therefore
Total volume (1+2+3+4+5+...+n): pi * h * (12+3 2+4 2+...+k 2) * r 2/k 3.
=pi*h*r^2* k*(k+ 1)*(2k+ 1)/6k^3
=pi*h*r^2*( 1+ 1/k)*(2+ 1/k)/6
Because the greater n is, the closer the total volume is to the cone volume, and the closer 1/k is to 0.
So pi * h * R2 * (1+1/k) * (2+1/k)/6 = pi * h * R2/3.
Because v-pillar = pi * h * r 2
therefore
The V-cone is 1/3 of the volume of a V-column with equal bottom and equal height.