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endless and the search for beauty and recursion

The other day I suggested a way of showing that the sum of the first odd numbers is always a perfect square. Use chickpeas, remember?
In mathematics, sometimes the obvious is not always easy. When faced with a problem, often, the hardest part is selecting the right tool. Once this is chosen (well chosen, that is), the resolution becomes trivial.
Today I propose a problem to solve using a common tool but most of the fans do not know math: the ralación between algebra, geometry and arithmetic.
The approach is simple: How much is this infinite sum?
adding What is the infinite things that, as the mathematicians love. If you want to try on their own to stop reading and get back in time with the solution (1 / 3, in case you are curious and want to stop reading right now).

methods exist for adding this series easily, anyone who has made a first course in a Bachelor of Science known, but since this is a blog for men of letters I I save that way.

The first impulse is brute force: to make all these fractions in decimal numbers and put them into a calculator (better in Excel) to see what comes out. The sum of the top five is: 0.25

+0'0625 +0'015625 +0'00390625 +0,00390625 = 0.3330078125

if we add fractions, we see that the first decimal always 3, so if one is sufficiently clever, it can venture and hit the correct solution (1 / 3 = 0'333333 ...)

But in mathematics, as in justice intuition is not enough to solve. It must be demonstrated.
And brute force can be very effective, but is not elegant, and mathematics, as important as the rigor is the pursuit of beauty: Look at this picture

:
is a triangle. Let's put another triangle inside it:
the triangle we have divided into 4 equal parts (what he seems to have put a thong?). We are left with the central and painted blue. Blue triangle that represents a quarter of the large triangle.

We'll repeat the process in the upper triangle:
That blue triangle above represents a quarter of a quarter of the big triangle, that is 1 / 4 ²
can repeat as many times as you want:
To sum just have to keep in mind that each tiángulo blue is the fourth of which is directly below. Therefore, the blue triangles chain represents the infinite sum we want to calculate.
Now watch the original triangle has been divided into three pieces exactly the same (I have painted different colors to distinguish them), and if the three pieces are identical, then each of them represents 1 / 3 of the triangle.

Therefore our infinite sum of fractions (or triangles) is 1 / 3.
You see, math is not enough that all things are correct, it is also important that they are beautiful. I know that for one who had bad experiences in mathematics at school, talking about beauty in a problem is almost a provocation, but I do not deny that, at least, is smart. It seems to me wonderful.