In this chapter, we present the Cauchy condensation test (named by Augustin Louis Cauchy). It allows us to only check the condensed series
for convergence , which contains way less elements than the original series
. More precisely, in case the series elements
are non-negative and decreasing, we know that the original series
converges if and only if the condensed series
converges.
The derivation will involve a direct comparison to a divergent harmonic series
and convergence to a convergent generalized harmonic series
für
.
Repetition and derivation of the criterion[Bearbeiten]
For proving divergence of the harmonic series
, we used a lower bound for each
-th partial sum
:
How can we generalize this concept to a general series
? In order to make the same estimation steps, the series must have some properties identical to the harmonic series:
- The series elements
have to be non-negative.
- The sequence of elements
has to be monotonously decreasing.
If
is a series with non-negative elements fulfilling
for all
. Then
So we estimated
from below by
. That means, that if the condensed series
diverges and hence,
as well, then the series
also diverges by direct comparison. Conversely (by contraposition), if
converges, then
converges, as well.
For the proof that the generalized harmonic series
with
converges, we compared the
-th partial sum
for
to a convergent geometric series
. The bounding worked as follows:
We try to do the same for a general series
with
- non-negative elements

- and
for all
(monotonously decreasing sequence of elements)
Let
. then,
So we can also bound
from above by
. Direct comparison can again be applied and leads us to the conclusion: If the condensed series
converges, the the original series
also converges.
So if elements are non-negative and monotonously decreasing, we have an equivalence between the convergence of the series
and the condensed series
. This result is called Cauchy condensation test. It can be very useful, to remove logarithms out of a series. For instance, if
. Then, for the condensed series
. So condensation can remove double logarithms.
Now, let us formulate these findings in a mathematical language, i.e. a theorem with a proof:
Theorem (Cauchy condensation criterion)
Let
be a non-negative and monotonously decreasing sequence. Then, the series
converges if and only if the condensed series
converges.
Proof (Cauchy condensation criterion)
Proof step: "
"
Proof step: "
"
Hint
Analogously, one can show that a non-positive series with elements monotonously increasing (e.g. up to 0) converges, if and only if the condensed series converges.
Warning
The condensation test is NOT treated in all calculus courses. The main reason for this is that there are not too many applications, where it is useful. Basically only for eliminating logarithms. Please, only quote this test in your exercise solution, if it was treated somewhere in the lecture! Otherwise, you may reduce the condensation test to a direct comparison test, by making the same estimates as in the proof above.
Exercise (Removing logarithms)
Investigate for
whether the series
converges.
Proof (Removing logarithms)
By means of the Cauchy condensation test, this series converges if and only if the following condensed series converges:
This is just a generalized harmonic series with a constant pre-factor
. The example above shows that this series converges if and only if
and diverges if and only if
.