The null space in an eigenvector function can be summed in linear systems, by linear algebra. - is proof of this.
I am only showing it this way, so as to desribe the irresistible truth in Plato’s Meno-s slave,as a way to reach the anti derivative. Kind of like showing the wisdom of zeno’s Paradox, and it’ solution being solvable backwards.
Plato, it seems left it to the future to come up with an explanation of this idea, or of ideas in general.
A “converging series” specifies a LIMIT for a summation. The LIMIT is equal to xxx. You can have two obviously different summation series with the same limit. Note that the summation must REACH INFINITY before it can equal its limit. And there is no “at infinity” to be reached.
And even a higher truth is that even if the summation could actually reach infinity, it still would not equal its limit because its definition forbids it from EVER summing up the last tiny bit.
That is irrelevant and Zeno’s can be solved easier than that via relative infinitesimals, “calculus”.
That is what must happen in order to make 1.0 = 0.999…, but the problem is that such can never happen.
Strange! Before James and I posted one could read the following text below Jerkey’s last post, although he was the last one who posted (before James and me):
“Last edited by jerkey on Wed Jun 29, 2016 5:24 pm, edited 2 times in total.”
Yes. Therefore all this operations have to do with the infinitesimal calculus (inveted by Gottfried Wilhelm Leibniz). So James is right with his answer.
James, you’re treating an infinite string as though it’s just a really, really long finite string (thus leaving a “last little bit”, presumably after the “last digit” at the “end” of the “endless” string…). That isn’t true. A really, really long finite string has a last digit, an infinite string does not. A really,really long finite string would indeed leave a “last little bit” left over, but we aren’t talking about a really, really long finite string, we’re talking about an infinite string. That distinction is crucially important.
No. That is what YOU are doing when you claim that there is a 0 “at the end once it gets to infinity”.
I am saying that because there is no end, the 9s must still be going - ALWAYS (true by definition). And thus never, ever becoming the string of 0s required for “1.000…”.
In Your response ending in ‘that clould never happen’ -5 of Your responses back in the forum, and the one two back, ending with ‘impossible’ - referring to machine intelligence.
It could never happen - is equivocal to - impossible
The claim is that there are two decimal expansions for the number 1: a 1 with an infinite string of 0s after the decimal, and 0 with an infinite string of 9s after the decimal. Both are equivalent decimal expansions for the same integer. So there is no requirement that the 9s morph into a string of 0s, the string of 9s itself is equal to 1.
Multiple forms of the same value are common in math: 1/2 = 2/4 = 3/6 = 0.5 = .4999…