jsfisher said:
You also have a consistent failure in reading comprehension. Here, for example, the challenge was for you to show that X is 0 in the limit
The failure in reading comprehension is yours, in this case.
If we deal with an infinite convergent series, then exactly because of the
inseparable linkage between constant X AND (2a+2b+2c+2d+...) convergent series, X-(2a+2b+2c+2d+...) > 0, and as a result (2a+2b+2c+2d+...) < X.
In other words, approaches is a constant property of (2a+2b+2c+2d+...) infinite convergent series.
An infinite convergent series is a result of infinitely many points (where a point is a local atom because it can be simultaneously in exactly one location) and infinitely many segments (where a segment is a non-local atom because it can be simultaneously in at least two locations, and a segment is not made by points).
The value of a given limit is reached only if
finitely many points and segments are involved, but then we are not dealing anymore with an infinite convergent series.
My proof without words is rigours and clearly and simply shows the clear difference between a finite collection and non-convergent series of points and segments that indeed reaches the value of a given limit, and an infinite convergent series of points and segments that permanently approaches (does not reach) the value of a given limit.
Here is the proof without words, and its inevitable result about the clear difference between permanently approaches, and actually reaches:
1) Take a straight 1-dim with length X.
2) Bend it and get 4 equal sides along it.
3) Since the length between the opposite edges is changed to the sum of only 3 sides, and since the number of the sides after the first bending is 4 sides, we have to multiply the bended 1-dim element by 1/(the number of the bended sides), in order to get back length X.
As a result the bended 1-dim element has length X, but the length between its opposite edges becomes smaller (it converges).
In general, this convergent series of 1/(the number of the bended sides) is resulted by 1/1+1/4+1/16+1/64+1/256+... , where X is subtracted by (2a+2b+2c+2d+…)
Here is the result:
[qimg]http://farm5.static.flickr.com/4015/4430320710_daf5b36c0f_o.jpg[/qimg]
4) By Standard Math X – (2a+2b+2c+2d+…) = 0
5) (4) is false because (2a+2b+2c+2d+…) can be found as long as X is found.
6) Since X is found upon infinitely many scale levels then (2a+2b+2c+2d+…) must be < X , and as a result X - (2a+2b+2c+2d+…) > 0.
7) Conclusion: (2a+2b+2c+2d+…) does not have sum X.
A shorter version of the proof:
X = the constant length > 0
[qimg]http://farm5.static.flickr.com/4015/4430320710_daf5b36c0f_o.jpg[/qimg]
1) By Standard Math X – (2a+2b+2c+2d+…) = 0
2) (1) is false because (2a+2b+2c+2d+…) can be found as long as X is found.
3) Since X is found upon infinitely many scale levels then (2a+2b+2c+2d+…) must be < X , and as a result X - (2a+2b+2c+2d+…) > 0.
4) Conclusion: (2a+2b+2c+2d+…) does not have sum X.
The Man, jsfisher and sympathic, this result is logically irresistible truth.