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

[Applejacks]

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?

 

[vela]

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.

 

[Applejacks]

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])?

 

[micromass]

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])?

Yes. They are only formal power series. When looking at $R[[X]]$, we do not care about convergence. So $\sum n!X^n$ is a perfectly valid element of $R[[X]]$.

 

[Hurkyl]

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

$$(1 - X)^{-1} = 1 + X + X^2 + X^3 + X^4 + \cdots = \sum_{i=0}^{\infty} X^i$$

You can check this fact by multiplying the right hand side by 1-X.