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Polynomial Function Definition Question

  1. Nov 2, 2005 #1
    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...

    [tex]\sum_{n = 1}^{\infty} x^{2n-1}(-1)^{n+1}/(2n-1)![/tex]
    Last edited: Nov 2, 2005
  2. jcsd
  3. Nov 2, 2005 #2


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    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.
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