Given a Taylor series about a point
for a function
where
and
we can find the Taylor series about
for
by putting
and equating powers of
in the identity
to obtain equations for
etc in terms of
Since we know the
we can evaluate the
Example: Obtain the Taylor series for the inverse of
about
The Taylor series for
about
is
where
and
satisfy
Equating coefficients of powers of z- %apha =z we obtain
Then
Since labels are arbitrary, we can write