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Disproof to the Pythagorean Theorem?

 Quote by D H A series all of whose elements have the same value is a convergent series.
Yes, let me state it how I intended, It is not converging to the length/limit of the hypotenuse. Each iteration of the series does not have a "delta", it is constant, most series with a limit, exhibit a property of sums approaching the limit. (I'm just not eloquent, I do know what I mean)
 I don't understand why you would expect a+b to approach the length of the hypotenuse, or change at all. No matter how tiny you make them or how many of them their are, or how smooth the drawing looks to your eye, so long as there isn't a diagonal line involved, you haven't created a triangle. You're basically asking why the value of $\frac{1}{n} (an + bn)$ never changes no matter what you stuff into n.
 Can't this question be generalized as: what function defines the distance between two points on a plane and what simplified version of this function arises from minimising its value ?

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 Quote by justsomeguy I don't understand why you would expect a+b to approach the length of the hypotenuse, or change at all.
Naively, one might expect that if a sequence of functions {fn(x)} converges on some function f(x), then the sequence {O(fn(x))} formed by applying some operator O to each element of that sequence of functions will converge on the result of applying that operator applied to that limit function f(x).

That's an invalid expectation. This supposed disproof of the Pythagorean theorem demonstrates that this is invalid.
 This is Hilbert's Manhattan Metric, sometimes called Taxicab Geometry. Wikipedia... Taxicab Geometry
 Well I think the users before me answered the question pretty well. (The sequence of derivatives not converging uniformly to the diagonal's slope is the issue) A similar problem I noted before is the formula for length of a curve in polar coordinates. We all know that the formula for the area of a curve in polar coordinates can be found by separating the area into small sectors of angle dθ and approximating as triangles or circular sectors (much like we use rectangles in cartesian integration). But the same method won't work to derive the length of a curve in polar coordinates (approximating the curve as small sectors of circles). One must instead use the cartesian formula and convert into polar. (that's at least what I've learned, if there's a different way I'd like to see it)

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