## 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

is called a power series centered at , or a power series about ''.

Example 6.5.5   Consider

 equality valid when

Here conceptually we are treating like we treated 0 before.

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

 converges for all

Notice that the second series converges for , whereas the first converges only for , which isn't nearly as good.

William Stein 2006-03-15