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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.

Sedumjoy
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  • A polynomial has a finite number of terms (monomials). A power series can have a monomial in each degree. Example: the geometric series: $1+x+x^2+\dots+x^n+\dots=\sum_{n\ge 0}x^n$. – Bernard Jan 21 '18 at 20:41
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    although power series are not polynomials Read this more carefully. It does not say or imply that polynomials are not power series, which your title seems to be asking. – dxiv Jan 21 '18 at 20:44
  • you are correct. it should be in what sense is a power series not a polynomial ...... – Sedumjoy Jan 21 '18 at 20:53
  • I am confusing polynomial vs.polynomial function. and also if a power series does not converge then perhaps it cannot be defined as a function so there you have a case that the power series is not even a function. – Sedumjoy Jan 21 '18 at 20:56
  • Nowhere do they say that "a polynomial [is] not a power series" so, why the title (and the whole post, actually)? – Did Feb 13 '18 at 01:58

3 Answers3

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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.

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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?

szw1710
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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.

Mohammad Riazi-Kermani
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