![]() ![]() The commutative and associative properties do not hold for conditionally convergent series. The sum of its negative terms diverges to negative infinity. The sum of its positive terms diverges to positive infinity.ĥ. It has both positive and negative terms.Ĥ. The series is convergent, that is it approaches a finite sum.ģ. ![]() Their "proof" was utter nonsense.Ī series is defined to be conditionally convergent if and only if it meets ALL of these requirements:Ģ. Thus when S fails to exist, it is possible to get various nonsensical and contradictory solutions.īasic mathematical operations all require that S exists, if S does not exist the operations can still produce "answers" but they will be nonsense.īTW, the Numberphile video where they "proved" that S of the positive integers was -1/12 made use of such nonsense with divergent series. Since it is possible to have multiple contradictory sums for S, it must be the case that S fails to exist. And, for that matter, it does not hold that S + S = 2S. It is not the case that the associative property holds for this particular series. It Is NOT the case that S=½ nor any of the other values we could come up with. This is obviously absurd and self-contradictory. So let us say that I added S to itself 999 times, giving me 1000S = S. It is still 1-1+1-1+1-1.Īnd, of course, I can add as many sets of S to each other as a like and they will still be the same sum, they will still be 1-1+1-1+1-1. Notice that if I add another copy of the sum 1-1+1-1+1-1. They have no more meaning than the "proofs" 1=2 which contain a hidden division by zero. In short, S fails to exist for a divergent series, thus computations with S are meaningless. ![]() You cannot assume the associative property applies to an infinite series, because it may or may not hold. ![]()
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