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: ""
For the sequence of partial sums there is
If the sequence / series converges, then also the subsequence will converge. The above bounding implies that then also converges. Multiplying by , we also get convergence of the condensed series .
Proof step: ""
For the -th partial sum and for , there is:
now, if the series converges, then by the above estimate, must be bounded and hence convergent.
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.
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.
Example (Cauchy condensation test and the zeta function)
As seen above, the Cauchy condensation test can be used to prove the convergence / divergence for the generalized harmonic series for . This series converges if and only if the condensed series converges:
This is just a geometric series with . It converges if and only if
By the Cauchy condensation test, the generalized harmonic series converges if and only if . Especially, for , it diverges. Usually, in the convergent case , one defines the
Riemann zeta function as the limit of this generalized harmonic series .
You may just use in an exercise that this result is finite, without referring to the Cauchy condensation - or any other test especially,
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 .