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

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

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

Example 6.5.5   Consider

$$\sum_{n=0}^{\oo } (x-3)^n = 1 + (x-3) + (x-3)^2 + \cdots$$

$$= \frac{1}{1 - (x-3)} \vert x-3\vert<1$$

$$= \frac{1}{4-x}$$

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,

$$\frac{1}{4-x} = \frac{1}{4} \cdot \frac{1}{1-x/4}$$

$$= \frac{1}{4} \cdot \sum_{n=0}^{\oo } \left( \frac{x}{4}\right)^n \vert x\vert < 4 .$$

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.