What's the difference between Taylor series around different points? Besides of computational aspects what's the difference between a Taylor series around a point $\alpha$ and a Taylor series around a point $\beta$?
 A: The series may have different radii of convergence, for example. I assume you start with a reasonably nice function $f$. Suppose $f$ has a pole, I mean, you are "dividing by 0" somewhere, say, at $x=a$. If nothing else happens, then the radius of convergence at $\alpha$ cannot be larger than the distance from $a$ to $\alpha$, but it could be precisely this distance. Same with $\beta$. 
To be specific, take $f(x)=1/(1-x)$. The series at 0 is $\displaystyle \sum_n x^n$ which converges for $|x|<1$. The series at $-1$ is $\displaystyle \sum_n \frac1{2^{n+1}} (x+1)^n$; it converges for $|x+1|<2$. 
This explains why even in the case of reasonably well behaved functions we may want to compute the Taylor series at different points depending of the region we are interested in studying.
A: Suppose $f(z)=\sum_n a_n(z-z_0)^n$ on some open subset $S$ within the disc of convergence $B(z_0,\rho)$ (we could take the expansion to be valid on the whole disc too). Then given another $w \in S \subseteq B(z_0,\rho)$, we can find a $B(w,\delta) \subseteq S \subseteq B(z_0,\rho)$ such that $f$ can be expanded as a power series with $w$ as a center. i.e
$f(z)=\sum_{n=0}^{\infty}\hat{a}_n(z-w)^n$ valid on $B(w,\delta)$
Where $\hat{a}_n=\sum_{i=n}^{\infty}\binom{i}{n} a_i(w-z_0)^{n-i}$
[One use this to show that the derivative of $f$ at a point $w$ inside the disc of convergence is just the first coefficient of the expansion around that point.]
