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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?

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    $\begingroup$ I would personally be more inclined to use "power series" if I were talking about something more formal or algebraic, while I would use "Taylor series" in a more analysis-oriented setting. Of course, that doesn't mean it's correct. :) $\endgroup$ – Zach L. May 15 '13 at 22:22
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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.

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  • $\begingroup$ Hint about the theorem mentioned: Given smooth function $f(x)=h$ for $\newcommand\abs[1]{\lvert#1\rvert}\abs x\le1/(2\abs h+1)$, and $f(x)=0$ for $\abs x\ge2/(2\abs h+1)$, we have $f(0)=h$ and $f^{(n)}(0)=0$ for $n>0$. Integrate on $[0,x]$ $n$-times, we get a smooth function $F(x)$ such that $F^{(n)}(0)=h$ and $F^{(m)}(0)=0$ for $m\ne n$. Use $F$'s to construct a series of functions, and prove that any times of termwise derivative of the series converges uniformly, and therefore the desired smooth function comes out. I don't know whether there's an easier approach. $\endgroup$ – Yai0Phah Jun 16 '13 at 15:56
  • $\begingroup$ Shouldn't it be $\sum_{n=0}^{\infty} a_n x^n$ instead of $\sum_{n=1}^{\infty} a_n x^n$? $\endgroup$ – MrAP Dec 23 '17 at 6:27
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Taylor series are a special type of power series. A Taylor series has a very special form, given by $$T_f(x) = \sum_{n = 0}^{\infty}\frac{f^{(n)}(x_0)}{n!}(x-x_0)^n,$$ and a general power series looks like $$P(x) = \sum_{n = 0}^{\infty} a_n (x - x_0)^n,$$ where the $a_k$'s are just the constants associated to this power series in particular. The $a_n$'s may not have the form $f^{(n)}(x_0)/n!$, so that not every power series is a Taylor series (although every Taylor series is a power series). Edit: as Matt noted, in fact each power series is a Taylor series, but Taylor series are associated to a particular function, and if the $f$ associated to a given power series is not obvious, you will most likely see the series described as a "power series" rather than a "Taylor series."

Both of these types of series can be generalized to forms involving more variables, and you can also come up with types of series that involve negative powers of $x$.

As for generating functions, these are more formal objects, the analysis of which doesn't really deal with the issue of convergence as much as the analysis a power series or a Taylor series does. In this case, the coefficients are encoding information about some sequence of numbers $\{a_n\}$, and we examine the series formally to gather information about this sequence.

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    $\begingroup$ Dear Stahl, Of course, there is a theorem of Borel which says that every power series (with $a_n$ being real numbers, say) is the power series of some smooth function $f$ around the point $x_0$. Regards, $\endgroup$ – Matt E May 15 '13 at 22:30
  • $\begingroup$ Good to know; I wasn't familiar with this theorem! (I assume you mean each power series is the Taylor series of $f$ around $x_0$, yes?) $\endgroup$ – Stahl May 15 '13 at 22:32
  • $\begingroup$ Dear Stahl, Yes, that's what I mean. Cheers, $\endgroup$ – Matt E May 15 '13 at 22:36
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    $\begingroup$ Does the downvoter care to explain the reasoning? $\endgroup$ – Stahl May 15 '13 at 22:46
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    $\begingroup$ What's the difference between a "person" and "child?" Everyone is someone's child! I have two persons, aged six and nine. $\endgroup$ – Kaz May 16 '13 at 3:00
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A power series is a purely algebraic object, defined as a formal infinite sum $\sum_{n=0}^\infty a_nx^n$ where the $a_n$ are elements of some ring $R$ (for example, $R=\mathbb{R}$ or $\mathbb{C}$). You can choose any $a_n$ you like, and you still have a well-defined power series - one need not be concerned with questions of convergence. The set of all power series over a ring $R$ itself forms a ring $R[[x]]$. These power series do not in general define functions from $R$ to $R$; there is in general no way to make sense of an infinite sum of elements of $R$, so there is no sensible way to substitute an element of $R$ for $x$ in an arbitrary power series.

A Taylor series is a special kind of power series $T_{f,x_0}$ defined using a (real or complex) smooth function $f$ and a real/complex number $x_0$, as in Stahl's answer. Here we do have a sensible way to talk about convergence of limits in $\mathbb{R}$ and $\mathbb{C}$, and indeed we can interpret $T_f$ as giving us a function defined on a neighbourhood of $x_0$.

A generating function is a power series of the form $\sum_{n=0}^\infty a_nx^n$ where the coefficients $a_n$ are natural numbers. This is an algebraic object which encodes the sequence $\{a_n\}_{n=0}^\infty$ and does not in general define a real-valued function.

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    $\begingroup$ Not all power series are formal objects, often we have analytic power series just describing some function of a real or complex variable. A general power series (coefficients in $\Bbb R$ or $Bbb C$) will converge at least at the point it's centered around, although it might converge with a larger radius if the coefficients behave nicely. However, as a fan of algebra, I do appreciate the explanation of formal algebraic power series, as this hasn't really been explained in this thread yet. $\endgroup$ – Stahl May 15 '13 at 22:50
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A power series is just a series whose terms are monomials in some number of variables, such as $$ \sum_{n=0}^\infty a_n x^n \qquad \text{or} \qquad \sum_{m,n=0}^\infty a_{m,n} x^my^n. $$ These are sometimes formal algebraic objects that encode sequences in their coefficients.

In my experience, the term Taylor series is used when the power series is built from a function. In this context, issues of convergence are central.

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