4.3 Taylor–Expansion of
Functions
4.3.1 Convergence: Expansion of
The derivatives of this function are
We use this to expand
around
,
Now we ask: for which values of
does this Taylor series actually converge? We use the
ratio test and write
From Eq. (
4.14
), we recognise that the
series
- converges for
.
- diverges for
.
At
we don’t expect the series to converge because
is undefined (minus infinity). To decide
what happens at
, we have to invoke an additional convergence test:
Leibnitz’ test for alternating
series: The alternating series
converges, if
for all
, and
.
We apply this rule to the case
of our series Eq. (
??) for
: At
,
and
, that means the Leibnitz’ test tells us that the
series converges at
. The result gives us a famous formula for
,
and we summarise our results for
as
We say that the
radius of convergence
of this series is
. For values of
beyond that radius, the series diverges and does no
longer represent the function
. In other words, the Taylor series Eq. (
4.31
) is only useful for
‘small’
.
4.3.2 Alternative way to generate a Taylor
Series
Let us write
in a ‘complicated way’, i.e. as an
integral:
Now, we use our result for the geometric
series, Eq.(
4.33
),
(
‘LEARN THIS ONE BY
HEART’) with
, which leads to
We integrate this term by term, which is
easy,
which is the same as Eq. (
4.31
)
4.3.3 Taylor expansion of
around an arbitrary
So far we have always expanded our
functions
in the vicinity of
, i.e. ‘around’
:
Taylor expansion of
around
,
The Taylor expansion of a function
near
is performed in an analogous way, but with
replaced by
, and
replaced by
:
Taylor expansion of
around
,
In some books, the special case of a Taylor
series around
is called
Maclaurin Series.