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CSfilm, I don't think I've ever seen it stated that it's the circumference:diameter ratio of a circle on a flat plane. Not saying this isn't the understood definition of 3.14159..., but that technically you could define a different surface, pick a point, draw a circle around it, and calculate different values of pi.
Why is it that the sum of 4/{n[(-1)^(1+n)]} approaches pi (of the 3.14159... variety)?
This is the 4/1-4/2+4/3-4/4+4/5-4/6+4/7... series. It doesn't seem like there's any reason that it should.
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Simple Machines in Higher Dimensions
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