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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

    vela

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    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

    micromass

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    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

    Hurkyl

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    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.
     
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