I Polynomial of finite degree actually infinite degree?

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Hmm... I couldn't help thinking about this while I was watching "Alita Battle Angel" today ($15 for popcorn and soda, holy choo-choo!!).

I suspect everybody concurs that the expansion of 1/(1-x) in power series (an Analysis result) only makes sense if 0<x<1.

I think the point that Math_QED made on post #8 goes about this: if we set the series expansion aside, and only consider infinite polynomials, and say I define these purely Algebraic functions:

f:ℝ→ℝ, f(x) = 1+x+x2+x3+...
g:ℝ→ℝ, g(x) = x+x2+x3+...
h:ℝ→ℝ, h(x) = f(x) - g(x)

Then what's the value of h(2)? I think I empty-mindedly argued that, going by its definition, then h(2)=∞-∞ is undefined, (therefore h(x) only makes sense for 0<x<1), while Math_QED corrected me saying that if we replace f(x) and g(x) in the definition of h(x), and do the subtraction, then h(2) = 1 (therefore h(x) is valid for any x).

So, what's the truth of this matter? What's the value of h(2)?
 

Math_QED

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Hmm... I couldn't help thinking about this while I was watching "Alita Battle Angel" today ($15 for popcorn and soda, holy choo-choo!!).

I suspect everybody concurs that the expansion of 1/(1-x) in power series (an Analysis result) only makes sense if 0<x<1.

I think the point that Math_QED made on post #8 goes about this: if we set the series expansion aside, and only consider infinite polynomials, and say I define these purely Algebraic functions:

f:ℝ→ℝ, f(x) = 1+x+x2+x3+...
g:ℝ→ℝ, g(x) = x+x2+x3+...
h:ℝ→ℝ, h(x) = f(x) - g(x)

Then what's the value of h(2)? I think I empty-mindedly argued that, going by its definition, then h(2)=∞-∞ is undefined, (therefore h(x) only makes sense for 0<x<1), while Math_QED corrected me saying that if we replace f(x) and g(x) in the definition of h(x), and do the subtraction, then h(2) = 1 (therefore h(x) is valid for any x).

So, what's the truth of this matter? What's the value of h(2)?
The reason why you are so confused is that you are mixing up two concepts.

Consider expressions of the form ##\sum_{n=1}^\infty a_n X^n##. These are called "formal power series".

Now, there are two ways to go: the algebraic way or the analytic way.

In algebra, we just define an addition and a multiplication with these formal power series: we consider formal series as elements on which we define some abstract operations which give another formal series, i.e. a series of the form ##\sum_{n=1}^\infty a_n X^n##.

In analysis, we consider ##\sum_{n=1}^\infty a_n X^n## as a limit of partial sums

##i.e. \sum_{n=1}^\infty a_n X^n = \lim_{k \to \infty }\sum_{n=1}^k a_n X^n## and we can ask ourselves for what values of ##X## this limit makes sense (= exists). If all the limits exist and some conditions are fulfilled, the algebraic operations that we can make with series in this way coincide with the ones that are defined in the algebraic way, but this is not always the case.
 

FactChecker

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So one can logically talk about the algebraic equality of expressions but should be careful about any implications regarding their numerical evaluations.
 

Math_QED

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So one can logically talk about the algebraic equality of expressions but should be careful about any implications regarding their numerical evaluations.
That's a good TL:DR :)
 
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The reason why you are so confused is that you are mixing up two concepts.

Consider expressions of the form ##\sum_{n=1}^\infty a_n X^n##. These are called "formal power series".

Now, there are two ways to go: the algebraic way or the analytic way.

In algebra, we just define an addition and a multiplication with these formal power series: we consider formal series as elements on which we define some abstract operations which give another formal series, i.e. a series of the form ##\sum_{n=1}^\infty a_n X^n##.

In analysis, we consider ##\sum_{n=1}^\infty a_n X^n## as a limit of partial sums

##i.e. \sum_{n=1}^\infty a_n X^n = \lim_{k \to \infty }\sum_{n=1}^k a_n X^n## and we can ask ourselves for what values of ##X## this limit makes sense (= exists). If all the limits exist and some conditions are fulfilled, the algebraic operations that we can make with series in this way coincide with the ones that are defined in the algebraic way, but this is not always the case.
So, under the Algebraic understanding, is h(2)=1 or undefined?

I think that question is similar to this.. if we define f(x) = x/x, then is f(0)=1 or undefined?
 

Math_QED

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So, under the Algebraic understanding, is h(2)=1 or undefined?

I think that question is similar to this.. if we define f(x) = x/x, then is f(0)=1 or undefined?
I think your main confusion here is that you don't know the distinction between polynomials and polynomial functions.

A polynomial is a formal element ##\sum_{k=1}^n a_k X^k##.

A polynomial function is a map ##x \mapsto \sum_{k=1}^n a_k X^k##

You can evaluate a function in all points of its domain, so as for your question it does not make sense to ask what ##f(0)## is when ##f(x) = x/x## (here I regard ##f(x)## as a polynomial function), because ##0/0## is a meaningless expression. It is true however that ##f## has a continuous extension that is equal to ##1## everywhere.

Now, we can regard any polynomial as a polynomial function and then we have a notion of evaluating a polynomial. This can be generalised to general polynomial rings and should be treated in an algebraic geometry course.

Now, there is also the distinction between a formal power series and a power series.

A formal power series is a formal element ##\sum_{k=1}^\infty a_k X^k##.

A power series is a function ##x \mapsto \sum_{k=1}^\infty a_k x^k## with domain the elements ##x## where the limit of partial sums converges.

Unlike the polynomial part, I'm not aware of any algebraic evaluation of formal power series.

Thus, the question what is ##h(2)## makes no algebraic sense. Or at least not one that I'm aware of.
 
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I see; so that's what FactChecker referred to talking about equality of expressions, but being careful on the numerical evaluation of expressions!

That's a deep thought!! I'll have to sleep on it! This is fascinating! This means that two formula F(x) and G(x) may be algebraically equal, say in F(x) = x/x and G(x) = 1, one might think they are the same thing, but F(x=0) and G(x=0) may actually be different, wow!

Thanks for the explanations!
 

Math_QED

Homework Helper
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I see; so that's what FactChecker referred to talking about equality of expressions, but being careful on the numerical evaluation of expressions!

That's a deep thought!! I'll have to sleep on it! This is fascinating! This means that two formula F(x) and G(x) may be algebraically equal, say in F(x) = x/x and G(x) = 1, one might think they are the same thing, but F(x=0) and G(x=0) may actually be different, wow!

Thanks for the explanations!
Yes, in the fraction field of the polynomials ##X/X =1##.
 

Ray Vickson

Science Advisor
Homework Helper
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Hmm... I couldn't help thinking about this while I was watching "Alita Battle Angel" today ($15 for popcorn and soda, holy choo-choo!!).

I suspect everybody concurs that the expansion of 1/(1-x) in power series (an Analysis result) only makes sense if 0<x<1.

I think the point that Math_QED made on post #8 goes about this: if we set the series expansion aside, and only consider infinite polynomials, and say I define these purely Algebraic functions:

f:ℝ→ℝ, f(x) = 1+x+x2+x3+...
g:ℝ→ℝ, g(x) = x+x2+x3+...
h:ℝ→ℝ, h(x) = f(x) - g(x)

Then what's the value of h(2)? I think I empty-mindedly argued that, going by its definition, then h(2)=∞-∞ is undefined, (therefore h(x) only makes sense for 0<x<1), while Math_QED corrected me saying that if we replace f(x) and g(x) in the definition of h(x), and do the subtraction, then h(2) = 1 (therefore h(x) is valid for any x).

So, what's the truth of this matter? What's the value of h(2)?
You say "the expansion of 1/(1-x) in power series (an Analysis result) only makes sense if 0<x<1"

Actually, it makes sense if -1 < x < 1 (including x=0).
 
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thank you!
 

Stephen Tashi

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##1+x+x^2 = \dfrac{1-x^3}{1-x} = (1-x^3)\cdot \dfrac{1}{1-x} = (1-x^3)\sum_{k=0}^\infty x^k##.

Isn't this a contradiction since the LHS has degree ##2## while the RHS has infinite degree?
By the usual definition of "polynomial" the RHS is not a polynomial. (The usual definition of polynomial doesn't permit a "polynomial" to be specified by the operation of taking a limit. Do your course materials actually define a type of "polynomial" that has "infinite degree"? )

A polynomial function f(x) may be equal to a function that is expressed in a way not permitted in the definition of "polynomial". For example f(x) = x = x + sin(x) - sin(x).

What qualifies a function to be a polynomial is that it may be expressed in a certain way. Alternative ways of expressing it don't involve any logical contradiction.
 

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