Power Series Definition What does it mean for a series to be centered around a number? I'm taking complex analysis and am suddenly very confused. I didn't have this explanation, or proof of taylor and power series in calculus, and I'm thinking here, it grew out of complex analysis and not real. But, I'm lookin' through the book, tryin' to get at the proofs, and well, I can't seem to understand why the series is centered around a number. 
 A: A power series is by definition something of the form $\sum_{n=0}^{\infty} c_n (z - a)^n$ for some sequence of numbers $(c_n)_{n=0}^{\infty}$ and some number $a$.  It is said to be "centered at" $a$.
You can regard this as a purely formal thing (ie, just a word that goes with the data defining a power series).
You can also motivate the terminology as follows: if you want to think of the series just written as a function of the complex variable $z$, you need to know the domain of the function: the set of $z$ for which the series converges.  And it turns out that, no matter what the sequence $(c_n)_{n=0}^{\infty}$ is, the set of complex numbers $z$ for which the series $\sum_{n=0}^{\infty} c_n (z - a)^n$ converges is a disc centered at $a$.  So, power series "centered at $a$". 
(For this to make absolute sense and not just be a suggestive phrasing, you need to allow the single point $\{a\}$ and all of $\mathbb{C}$ as a "discs", and allow for junk on the boundary--- ie, not just "closed disc" or "open disc" but "open disc, plus perhaps some junk on the boundary"--- in your definition of "disc".  I hope this helped.)
