The product of two formal power series

and

is the formal power series
given by

where the coefficients
satisfy

for

If the leading coefficient
of
equal 1, then the above formula gives

Thus if the formal power series

satisfies

then its coefficients
can be computed with formula (), with the understanding
for
.
For instance, if
,
,
then

We shall show that if a formal power series
``converges'' on an interval, then the resulting function has derivatives of
all orders and its coefficients
are given by, exactly like the polynomials,

More is true: if
takes the form
then
for
.
This fact suggests a computational procedure to find
for a rational function
first, express
in the form
with
and
.
Notice that
cannot be zero, or else
would not be defined. Once
is factored out from
,
takes the form

Now the above method of expansion is applicable. Once the expansion

is found
is calculated from the formula

Example: Compute
for
in the expansion

Exercise:

(a) Find the power series expansion of up to

Let . Find for

The expansion

holds (the geometric series). Therefore, if
then
for all

Replacing
by
,
we have the expansion

Therefore, if
then
for all
.
To find the expansion for functions of the form
we write it in the form
to obtain the expansion

Therefore, if
then
.
By differentiating the geometric series on both sides, we obtain

Therefore, if
then
.
Again, differentiating this series we obtain

Therefore, if
then

Exercise:

Find out if . What if What if What if

If the rational function has the partial fractions decomposition in the form

the above method may be applied to find