A power series is a series of the form
where each is a number and
is a variable.
A power series defines a function
where we substitute numbers for
.
Note: The function is only defined for those
with
convergent.
For we computed
If is an infinitely differentiable function, then the
Taylor series of
at
is the series
Compute the Taylor series of at
.
We know
for all
. So
and
the Taylor series of
at
is
Compute the Taylor series at of
.
Write
. Then
,
,
,
.
In general,
so that
and the
Taylor series of
at
is
Given an infinitely differentiable function with Taylor series (at
)
either
converges
and is equal to
for every number
or
there is a number
(called the radius of convergence)
for which
converges and is equal to
for
while
diverges for
.
Differentiation: If
,
then
.
Integration: If
, then
.
Products: If
and
, then
.
Composition (monomial case): If
and
is a positive
integer, then
.
Compute the Taylor series at of
.
We know
. So,
.
Integrating,
.
That is,
.
Find the Taylor series at zero of
.
We know
so that
.
Multiplying,