TITLE: Are Taylor series and power series the same "thing"?
QUESTION [27 upvotes]: 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?

REPLY [2 votes]: 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:
$f(x) = \sum \limits_{n=0}^{\infty} \frac{f^{(n)}(0)}{n!}(x)^n$