By Leo Zippin
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The same type of argument applies also to series like 2 4 8 + + + . .. 5 125 25 Here a segment is divided into five parts, two are kept, two are worked on and one is given away ; we hold � of what is distributed and are led to an easy guess as to the answer. 21 . 21 as r + r2 + r3 + . . = ? Do you see how to get the right hand side in terms of r? 21, you will have a proof, by the method of Eudoxus-Archimedes, of the famous formula for the sum of all the terms of an infinite geometric progression whose ratio r is rational and smaller than j.
Proof? 23. ;7 . q, noticing that p - 2q is smaller than q . 24. Show that the irrationality of VB follows easily from that of v'2 . 25. Study p = yn . q given that there is an integer k such that kl < n < (k + 1)2. Show that if an integer is not the square of an integer, then it is also not the square of a fraction. 26. Show that 2V8 is an integer only if v'8 is an integer. 11 Computing the Square Root of Two Let us conclude this chapter with two inductive constructions for approximations to V2 .
But these two lines whose intersection should have given us the point X coincide ! Therefore the point X cannot be defined in the same way as the other points of the curve. One sees from the entire curve, once drawn, that it ought to be easy to locate the point X on CB ; but the previous definition of the quadratrix does not apply to this point. The same difficulty arises when one goes over to the analytic equa tion, as we shall see below. 13 shows how the point X can be calculated as a limit of points of the rest of the curve.