Little confusion on my part. I am reading wiki to learn power series and it says " a polynomial can easily be expressed as a power series..." and "one can view power series as being like polynomials of infinite degree ..." and "although power series are not polynomials..." . The quotation marks are what I printed directly from wiki.
My confusion is in what sense are they not? Since I can take any polynomial and express it as power series.
On
Polynomials are power series with almost all coefficients vanishing. Have a look at exponential function: $$e^x=\sum_{n=0}^{\infty}\frac{x^n}{n!}.$$ Is it really a polynomial?
On
A polynomial of degree n stops at the $n^{th}$ power of x. For example a third degree polynomial is $$ P(x) = a_0 +a_1 x + a_2 x^2 + a_3 x^3 $$ A Power series on the other hand can go for ever.
For example the power series for $e^x$ is indeed an infinite series which does not stop at any degree.
A power series is not necessarily a polynomial unless from some point on all the coefficients are zero.
A polynomial can be considered a power series whose coefficients are zero from some point on.
A more correct sentence can reconcile these views: "although in general power series are not polynomials..."
All polynomials are (finite) power series, not all power series are polynomials.