4.2 Taylor–Series
One of the main motivations to investigate
infinite series is the desired to write arbitrary
functions
as polynomials of infinite degree, i.e.
For each fixed
, this is an infinite series of the form
with
. An important question is, e.g., how to determine the
coefficients
for a given function
, and to decide for which values of
the series for
does converge.
4.2.1 The Exponential Function
We already know one example for such a
series which is the exponential function
You have to remember this formula
throughout your whole life. This series converges for
arbitrary values of (complex or real)
since (ratio test!)
By use of the this
exponential series one
defines the famous
Euler number
4.2.2 Power Series for
and
We repeat our result for the series that
define
and
:
4.2.3 General Case
Now we treat the case of an arbitrary
function
The above equation means that we try to
represent the function by an ‘infinite’
polynomial. In the following, we assume that all derivatives of
, i.e.
,
,
, ... etc. exist. We write
Collecting all terms, we find the
Taylor expansion of
around
,
We define the truncated Taylor
series
The truncated Taylor series for finite
is often used as an approximation for the function
. For larger and larger
, we expect that this approximation of the function
by a polynomial of degree
becomes better and better, if the series converges, of
course. Let us look at an example to see how this works:
4.2.4 Example: The Exponential Function
We calculate the Taylor series of
around
. To do so, we have to calculate the derivatives
This is particularily simple because all the
derivatives of
are
. This means that
We recognise that the Taylor expansion of
just reproduces our old result, Eq. (
4.16
).
We can apply the ratio test to the series for
the exponential. WIth
, we find
for every fixed
. The Taylor series for the exponent thus converges for
all
.