Polynomial of finite degree actually infinite degree?

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SUMMARY

The discussion centers on the apparent contradiction between the polynomial equation \(1+x+x^2 = \frac{1-x^3}{1-x}\), where the left-hand side (LHS) is a polynomial of degree 2, while the right-hand side (RHS) is expressed as an infinite series. Participants clarify that the LHS and RHS are not equal as functions due to the undefined nature of the RHS at \(x=1\). The conversation emphasizes the distinction between formal power series and polynomial functions, highlighting that manipulations involving infinite series require careful consideration of convergence.

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  • #31
Mr Davis 97 said:
##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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