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newolder wrote:^ That groanful series is known as Cesaro convergent. This is when the averages of the partial sums in a divergent series tend to a constant - in this case 1/2. The explanation begins around the 19 minute mark in Mathlogger’s video above. Satisfaction not guaranteed.
Galactor wrote:newolder wrote:^ That groanful series is known as Cesaro convergent. This is when the averages of the partial sums in a divergent series tend to a constant - in this case 1/2. The explanation begins around the 19 minute mark in Mathlogger’s video above. Satisfaction not guaranteed.
Do you happen to know if there are practical applications of the series where the convergent result bears out somehow?
TopCat wrote:It's gems like this that keep me here, amidst all the dross of the current affairs and politics threads, which I have still found no easy way of suppressing. My thanks to the OP.
newolder wrote:Galactor wrote:newolder wrote:^ That groanful series is known as Cesaro convergent. This is when the averages of the partial sums in a divergent series tend to a constant - in this case 1/2. The explanation begins around the 19 minute mark in Mathlogger’s video above. Satisfaction not guaranteed.
Do you happen to know if there are practical applications of the series where the convergent result bears out somehow?
Taming divergent series seems to be Cesaro’s only function but this guy at Quora has a few words to add: https://www.quora.com/What-is-the-signi ... -summation
The quote from Abel is probably apt:
“The divergent series are the invention of the devil, and it is a shame to base on them any demonstration whatsoever."
Galactor wrote:newolder wrote:Galactor wrote:newolder wrote:^ That groanful series is known as Cesaro convergent. This is when the averages of the partial sums in a divergent series tend to a constant - in this case 1/2. The explanation begins around the 19 minute mark in Mathlogger’s video above. Satisfaction not guaranteed.
Do you happen to know if there are practical applications of the series where the convergent result bears out somehow?
Taming divergent series seems to be Cesaro’s only function but this guy at Quora has a few words to add: https://www.quora.com/What-is-the-signi ... -summation
The quote from Abel is probably apt:
“The divergent series are the invention of the devil, and it is a shame to base on them any demonstration whatsoever."
There are of course many mathematical constructions that appear abstract and beyond real application. But you only have to look at imaginary numbers and see what the imaginary resistance is, say, of a capacitor or an inductor, (their impedance) which we can calculate via the imaginary number i before you realise that abstraction in mathematics can "collapse" somehow to reality. Imaginary numbers explain real results.
The -1/12 universe depends upon this sequence and convergence which, I find, so undermines the calculation, but it would be more palatable if there were similar applications such as impedance from imaginary numbers.
As to where we go with the "understanding" that the answer to the universe and everything is not 42 but -1/12, you tell me.
Galactor wrote:It's a bit of a groan this one. The infinite series of -1 +1 -1 +1 ... converging to 1/2 is deeply dissatisfying. Or whatever it was ...
tuco wrote:Galactor wrote:It's a bit of a groan this one. The infinite series of -1 +1 -1 +1 ... converging to 1/2 is deeply dissatisfying. Or whatever it was ...
Indeed, about as satisfying as square root of negative one as mentioned in one of the vids or perhaps even the three fisherwomen.
tuco wrote:Correct By demonstration I meant how to arrive to solution, but "fiddle around with numbers" is sufficient.
Any other solutions, anyone?
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