I was just wondering in the lingo of Mathematics, are these two "ideas" the same? I know we have Taylor series, and their specialisation the Maclaurin series, but are power series a more general concept? How does either/all of these ideas relate to generating functions?
Real Analysis – Are Taylor Series and Power Series the Same?
complex-analysispower seriesreal-analysistaylor expansion
Best Answer
As others have noted, a power series is a series $\sum_{n=1}^{\infty} a_n x^n$ (or sometimes with $x$ translated by some $x_0$, to become $(x - x_0)$). Normally when one says Taylor series, one means the Taylor series of some particular smooth function $f$. (So in mathematical speech, one wouldn't usually say "consider a Taylor series". You might say "consider a power series", or "consider the Taylor series of the function $f$". At least, this is my experience.)
One complication in making too much of a distinction is that any power series (say with real coefficients) is the Taylor series of a smooth function (this is a theorem of Borel). So the distinction is more terminological than logical.
Added: Borel's theorem is discussed here.