Are Taylor series and power series the same "thing"?

Solution 1:

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.

Solution 2:

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.

Solution 3:

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.

Solution 4:

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.

Solution 5:

• Power's series is a series of the form:

  $f(x) = \sum \limits_{n=0}^{\infty} a_n (x - a)^n$

• Taylor's series is a special case of the Power's series where:

  $a_n = \frac{f^{(n)}(a)}{n!}$

  giving the taylor's series the form:

  $f(x) = \sum \limits_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!}(x-a)^n$

• Maclaurin's series is a Taylor's series where a=0, and it has the form:

  $f(x) = \sum \limits_{n=0}^{\infty} \frac{f^{(n)}(0)}{n!}(x)^n$