Polynomial Function Definition Question

[Quadratic]

I was wondering if a converging infinite series which includes "x" to ascending whole number powers would fit under the definition of a polynomial function. As an example:

f(x) = [n = 1 -> infinity] Sum(x^(2n-1)(-1)^(n+1)/(2n-1)!)

Also written as:

f(x) = x - x^3/3! + x^5/5! - x^7/7! + x^9/9! ...

Since x stays out of the denominator, and since it will always be to a power of a whole number, I was wondering if it could technically be defined as a polynomial function. It sounds like it can be described as polynomial, but my only concern is that it goes on to infinity, which isn't really a number. Also, I'm not sure if it's proper for polynomials to converge to a number, or to be expressed in sigma notation. (That series converges at Sin(x), where x is an angular measure in radians, btw)

edit: I meant to put this in general math, but I seem to have pressed the wrong button, and I'm not sure how to delete/move it. :/

edit: testing out that LaTeX code thingy...

$$\sum_{n = 1}^{\infty} x^{2n-1}(-1)^{n+1}/(2n-1)!$$

 

[Hurkyl]

This is actually a good spot for this topic, since infinite sums are (usually) a topic to be addressed with analysis.


It's fine for polynomial to be expressed in sigma notation. For example, the following is a polynomial in x:

$$\sum_{k = 0}^{n} c_k x^k$$

and if only finitely many of the c's are nonzero, then the following is also a polynomial in x:

$$\sum_{k = 0}^{+\infty} c_k x^k$$


However, when you have infinitely many (nonzero) terms it is no longer a polynomial. We call such a thing a power series. Amidst the jungle of all possible functions, power series are among the closest neighbors to polynomials, but alas, they aren't actually polynomials.

 

[tacman]

A polynomial function is defined as a mathematical expression that can be written in the form of a polynomial, which is a sum of terms with non-negative integer exponents. In the example given, the function does fit this definition as it can be rewritten in the form of a polynomial by expanding the series. However, as you mentioned, the series goes on to infinity which means it is not a finite polynomial. Infinite series are not considered polynomial functions, but they can be approximated by polynomial functions.

In terms of convergence, it is possible for a polynomial function to converge to a number, but the series in the example is not a polynomial function. It is important to note that a polynomial function is a specific type of function and not all functions can be classified as polynomials.

In summary, the series in the example is not a polynomial function, but it can be approximated by a polynomial function and it does fit the definition of a polynomial. Additionally, the use of sigma notation does not change the fact that the function is a polynomial.