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May 30th 2014, 18:34:06
the -1/12 thing only works for a certain type of infinity which is used in some cases in physics. But there are infinite types of infinity (as proven by Cantor). When using the concept in math in general it has to satisfy infinity over all types. Harmonic series where the absolute value of the terms do not get successively smaller do *not* converge on this basis. When we use the "infinity" symbole in most cases we really mean limit as x approaches infinity, especially in real analysis. This is not adding infinite terms, but rather what result the function/series/whatever gets closer and closer to the larger it gets.
1-1+1-1+1.... is not divergent, it's undefined. There's a huge difference. Just like what happens in lim x->infinity of sin (x). It doesn't get successfully larger, but it also doesn't get closer to any particular answer.
For those of you that know about statics, there are different kinds of convergeance. For example if u is a uniform random variable between 0 and 1, then 1 - u has the same distribution (converges in distribution). (Functionally they converge). However it doesn't converge in probability or absolutely. While for simulation purposes this really doesn't matter, for other applications it certainly can.
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