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Organic
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This is the reason why I used the word "General".
Please show us how the limit concept is rigorous.
Please show us how the limit concept is rigorous.
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Please tell me if this is a correct picture of this definition:for real numbers we say that a sequence x_n indexed by the natural numbers tends to a limit x if for any e>0 there is an m in N such that |x_n-x| < e for all n>m.
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- e - e
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- |x_m-x| <---> - m<n --> |x_n-x|
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| - <---------'
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- 0 - 0
Yes, |x_n-x| is for (what you call) all points and also |x_m-x| and e.Perhaps if by that you mean that if you plotted all the points |x_n-x| on the real axis, then all of the ones you plot after the m'th lie in the interval [0,e]
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- e - e
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- |x_m-x| <---> - m<n --> |x_n-x|
| | |
| - <---------'
| |
- 0 - 0
This is the invariant state that for any given n there is m<n in N....for any e>0 there is an m in N such that |x_n-x| < e for all n>m.
|x_n-x| is for (what you call) all points and also |x_m-x| and e.for real numbers we say that a sequence x_n indexed by the natural numbers tends to a limit x if for any e>0 there is an m in N such that |x_n-x| < e for all n>m.
| |
- e - e
| |
- |x_m-x| <---> - m<n --> |x_n-x|
| | |
| - <---------'
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- 0 - 0 is the limit x in this case
Organic said:This is the invariant state that for any given n there is m<n in N.
Right?
It is a non-linear sequence of x1, x2, x3, ... that converges very slowly to the limit and run away from the limit (diverges to infinity) very rapidly, as we can see here: http://phys23p.sl.psu.edu/~mrg3/mathanim/calc_I/Newtons.htmlGenerally N-R converges very rapidly as can generally be indicated graphically
No, it is possible only if a curve become a straight line before the limit point, or changes its direction before or in the limit point.The rest of what you write is wrong: it is perfectly possible for a sequnce in N-R to attain the limit in a finite number of steps
So, can we define a rigorous definition to a limit of a sequence without using an epsilon?
When these words are used, do we can understand that no element in the sequence reaching the limit?...neighborhoods or nearness.
The rest of what you write is wrong: it is perfectly possible for a sequnce in N-R to attain the limit in a finite number of steps
Organic said:Hi Hurkyl,
When these words are used, do we can understand that no element in the sequence reaching the limit?
When these words are used, do we can understand that no element in the sequence reaching the limit?
I mean that the tangent line stays in one and only ony side of the curve, when N-R is used.What do you mean by 'change direction of curvature'?
I mean that the tangent line stays in one and only ony side of the curve, when N-R is used.What do you mean by 'change direction of curvature'?
Organic said:Matt,
In this case ( http://phys23p.sl.psu.edu/~mrg3/mathanim/calc_I/Newtons.html )
I am talking about a curve that has non-zero curvature at all points, and this non-zero curvature does not changes its direction or become zero curvature at or before the limit point.
Well why don't you actually say what you mean rather than just writing nonsense. Half the problems with your posts is you say one thing and mean something else all together, another problem is you don't actually seem to be proving anything...Organic said:I mean that the tangent line stays in one and only ony side of the curve, when N-R is used.
I'll write it again:
In this case ( http://phys23p.sl.psu.edu/~mrg3/mathanim/calc_I/Newtons.html )
I am talking about a curve that has non-zero curvature at all points, and this non-zero curvature does not changes its direction or become zero curvature at or before the limit point.