The Geometric Progression Summation... and a block of ice

How do you sum the following numbers?:
1, 1/3, 1/9, 1/27, ... 1/ 3^inf
or even
1, 1/3.14, 1/3.14^2, ...., 1/3.14^ inf?

Answer: Go check up the GP summation formula. It's like... a/1-r (quote a certain math pro). Might be wrong because I typoed, but whatever. The actual derivation can be found online, but I don't really want to talk about it here, so... now comes the block of ice.

Imagine you have a block of ice, a cup of lemonade and a person who sort of fails at physics/math/common sense. So the cup is a perfect cylinder. The top of the liquid surface (this lemonade has no surface tension w.r.t air!) has a surface area of 1dm^2. Now this fail person lowers in a (cubical) block of ice of dimensions r dm by 1 dm by 23452345345 dm with the face that has r dm^2 surface area in first.

Soon, he has lowered the ice block exactly 1 dm into the original water level. By nearly obvious, the liquid would rise by r/1-r dm, since r dm ^3 of lemonade is displaced and the surface area as it rises is 1-r dm^2 (lemonade can't seep through ice). That is also equivalent to ( 1/ r-1 ) - 1.

However, the not-really-that-good-at-physics/math/common sense person tries to find the level the water has risen through this method. First, he takes the displaced amount of lemonade and see how far it would rise without the ice. Then he takes the volume the ice has displaced with the newly risen water level and with that new displaced volume compares how much more the water level would rise again... (not being very clear here, am I? It's ok. I fail)

Anyway, I hope it is somewhat clear that the water level will rise by r+r^2+r^3+...
=( 1/ r-1 ) - 1

and 0<=r<1. Why this bound? Because if r<0, you're possibly working with anti-matter and I don't want to talk to you, and if r>1, you're probably at odds with murderous potatoes and as such are too dangerous to be with. No really, anybody who tries to compress a solid is scary.

So... how to end off? Right. The usual statement. Whatever algebraic manipulations left are left to the reader. That's right. And if any careless mistakes have been made please inform me, but do not expect anything to change (apart from the post being torn apart).

"In exams you have to be soft because the questions are hard."

EDIT!!!: formula is like a/(1-r). Aha! ok thanks to RM-Sanctus, whoever that is.
EDIT!!!!: *tear*