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putting the geometric back into geometetric series

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kamil9876:
Let OA=1 and OP is perpendicular to the big hypotenuse. In fact all the little hypotenuses are perpendicular to the big hypotenuse. Let .

One can now show with simple trigonometry and geometry that a consequence of this is that . Because the geometrical nature is recursive(ie the pattern of the triangles keeps repeating infinitely) one can show that the base of the next triangle is and the next one is etc....

Therefore the length of the base of the big triangle is equal to the sum of the little bases:



With geometry one may now work out that: and that the big hypotenuse is: .

Using Pythagoras' theorem on the big triangle we get:






therefore:


letting just so that it's clear.

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