1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Difference between R[[X]] and R[x]?

  1. Oct 3, 2012 #1
    Not so much a question. Rather, I don't quite understand the concept.

    R[[X]] is the formal power series a0 + a1x + a2x^2....
    R[X] consists of all elements in R[[X]] which have only finitely many non-zero coefficients.

    Can someone give me an example? Would, say, two different elements of R[[X]] be 1+x and 3+2x+x^3 and hence both of these polynomials would be in R[X]? When does R[[X]] ever have infinitely many non-zero coefficients?
  2. jcsd
  3. Oct 3, 2012 #2


    User Avatar
    Staff Emeritus
    Science Advisor
    Homework Helper
    Education Advisor

    How about something like
    $$f(X)=\sum_{n=0}^\infty \frac{X^n}{n!}?$$ It would be an element of R[[X]] but not R[X] as it's clearly not a polynomial.
  4. Oct 3, 2012 #3
    Ok that clears it up. I wasn't too sure if we could take a formal power series to n=infinity, which leads me to another question. If the n! was in the numerator instead, for the example you provided,would we get the same result (element of R[[X]] but not R[X])?
  5. Oct 3, 2012 #4


    User Avatar
    Staff Emeritus
    Science Advisor
    Education Advisor
    2016 Award

    Yes. They are only formal power series. When looking at [itex]R[[X]][/itex], we do not care about convergence. So [itex]\sum n!X^n[/itex] is a perfectly valid element of [itex]R[[X]][/itex].
  6. Oct 3, 2012 #5


    User Avatar
    Staff Emeritus
    Science Advisor
    Gold Member

    One interesting feature is that [itex](1 - X)^{-1}[/itex] is an element of [itex]R[[x]][/itex]. In particular,

    [tex](1 - X)^{-1} = 1 + X + X^2 + X^3 + X^4 + \cdots = \sum_{i=0}^{\infty} X^i[/tex]

    You can check this fact by multiplying the right hand side by 1-X.
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Similar Discussions: Difference between R[[X]] and R[x]?
  1. Int x' dV/|r-r'| ? (Replies: 0)

  2. R[x] UFD, then R UFD (Replies: 2)

  3. R^n\{x} is connected (Replies: 2)