Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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


    User Avatar
    Staff Emeritus
    Science Advisor
    Gold Member

    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.
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook