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Old 2020-04-02, 11:03   #18
Nick
 
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Dec 2012
The Netherlands

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This game does not get anywhere useful with series.
A similar game with polynomials and rational functions, however, is more fruitful.

For polynomials f and g with real coefficients, define f to be greater than g if f(x)>g(x) for all sufficiently large x.
To put this precisely, f>g iff there exists a real number c such that, for all real x≥c, f(x)>g(x).
Convince yourself of the following:
  1. For constant polynomials, this definition agrees with our existing definition of "greater than" for real numbers.
  2. For any f,g exactly one of the following holds: f>g, f=g,, g>f.
  3. For any f,g,h, if f>g and g>h then f>h.
  4. For any f,g,h, if f>g then f+h>g+h.
  5. For any f,g, if f>0 and g>0 then fg>0.
In technical terms, we summarize this by saying that the set \(\mathbb{R}[X]\) of all polynomials with real coefficients forms an ordered ring with ordering defined in the above way.

Given polynomials f,g with real coefficients, if g≠0 then we can form a fraction \(\frac{f}{g}\). This is called a rational function over the reals.
For rational functions \(\frac{f_1}{g_1}\) and \(\frac{f_2}{g_2}\), we define
\(\frac{f_1}{g_1}\) to be greater than \(\frac{f_2}{g_2}\) if \(f_1g_2>f_2g_1\) (using the ordering for polynomials defined earlier).
(Just as with fractions of integers, it is possible to express each rational function in more than one way, but it is not difficult to show that this definition is independent of the representations we choose.)
Convince yourself of the following:
  1. For rational functions with 1 as the denominator, this definition agrees with the earlier one.
  2. For any \(f_1,g_1,f_2,g_2\), exactly one of the following holds: \(\frac{f_1}{g_1}>\frac{f_2}{g_2}\), \(\frac{f_1}{g_1}=\frac{f_2}{g_2}\), \(\frac{f_2}{g_2}>\frac{f_1}{g_1}\).
  3. For any \(f_1,g_1,f_2,g_2,f_3,g_3\), if \(\frac{f_1}{g_1}>\frac{f_2}{g_2}\) and \(\frac{f_2}{g_2}>\frac{f_3}{g_3}\) then \(\frac{f_1}{g_1}>\frac{f_3}{g_3}\).
  4. For any \(f_1,g_1,f_2,g_2,f_3,g_3\), if \(\frac{f_1}{g_1}>\frac{f_2}{g_2}\) then \(\frac{f_1}{g_1}+\frac{f_3}{g_3}>\frac{f_2}{g_2}+\frac{f_3}{g_3}\).
  5. For any \(f_1,g_1,f_2,g_2\), if \(\frac{f_1}{g_1}>\frac{0}{1}\) and \(\frac{f_2}{g_2}>\frac{0}{1}\) then \(\frac{f_1}{g_1}\frac{f_2}{g_2}>\frac{0}{1}\).
In technical terms, we summarize this by saying that the set \(\mathbb{R}(X)\) of all rational functions over the reals forms an ordered field with ordering defined in the above way.
But if f is just the polynomial X then, for any real number c we have f>c.
So this ordered field contains a copy of the real numbers as a bounded subset!
It follows that this ordered field is not complete, and therefore that the usual rules of limits and calculus no longer apply here.
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