To pause this video and try this on your ownīefore I'm about to explain it. Larger and larger, that the value of our sequence You to think about is whether these sequencesĬonverge just means, as n gets larger and In other words, a conditionally convergent series has the property that you get different answers if you rearrange or regroup the terms.įour different sequences here. Thus, conditionally convergent series are quite difficult to work with and it is very easy to get nonsense answers that look like they are correct. Thus, it is possible (by using the associative property and/or the commutative property) to group the terms of a conditionally convergent series to make it look like the series converges to any arbitrarily chosen number or to make it look like the the series diverges. 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. It took 9 steps to reach 1.You cannot assume the associative property applies to an infinite series, because it may or may not hold. The conjecture asserts that, regardless of the value of n, the sequence will always reach 1, at which point we enter a continual loop ranging from 1 to 4, to 2 and back to 1.įor example, if n = 12, we have the following sequence: 12, 6, 3, 10, 5, 16, 8, 4, 2, 1. If the previous term is odd, the next term will be 3 times the previous term plus 1 (3n+1). If the previous term is even, the next term will be half the previous term (n/2). The Collatz mathematical conjecture asserts that each term in a sequence starting with any positive integer n, is obtained from the previous term in the following way: The Collatz conjecture, which is also referred to as the Ulam conjecture, Kakutani's problem, the 3n + 1 conjecture, Hasse's algorithm, the Thwaites conjecture, or the Syracuse problem, involves a sequence of numbers known as wondrous numbers or hailstone numbers. It is named after a mathematician named Lothar Collatz, who first introduced the concept in 1937, two years after completing his PhD. The Collatz conjecture is widely regarded as one of the unsolved problems in mathematics.
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