Shift the Origin

It is often useful to shift the origin of a power series, i.e., consider a power series expanded about a different point.

Definition 6.5.4   The series

$\displaystyle \sum_{n=0}^{\oo } c_n (x-a)^n = c_0 + c_1(x-a) + c_2(x-a)^2 + \cdots
$

is called a power series centered at $ x=a$, or ``a power series about $ x=a$''.

Example 6.5.5   Consider

$\displaystyle \sum_{n=0}^{\oo } (x-3)^n$ $\displaystyle = 1 + (x-3) + (x-3)^2 + \cdots$    
  $\displaystyle = \frac{1}{1 - (x-3)}$   equality valid when $ \vert x-3\vert<1$    
  $\displaystyle = \frac{1}{4-x}$    

Here conceptually we are treating $ 3$ like we treated 0 before.

Power series can be written in different ways, which have different advantages and disadvantages. For example,

$\displaystyle \frac{1}{4-x}$ $\displaystyle = \frac{1}{4} \cdot \frac{1}{1-x/4}$    
  $\displaystyle = \frac{1}{4} \cdot \sum_{n=0}^{\oo } \left( \frac{x}{4}\right)^n$   converges for all $ \vert x\vert < 4$$\displaystyle .
$    

Notice that the second series converges for $ \vert x\vert < 4$, whereas the first converges only for $ \vert x-3\vert<1$, which isn't nearly as good.

William Stein 2006-03-15