Cauchy condensation test

The Cauchy compression criterion, also known as Cauchy compression rate, compression principle, dilution rate or condensation criterion (after Augustin Louis Cauchy ), is a mathematical convergence criterion, ie a means of deciding whether an infinite series is convergent or divergent.

Formulation

Be a monotonically decreasing sequence of non-negative real numbers. Then, the infinite series

The same convergence behavior as the compressed row

That is, a series of accurately converges when the other converges.

Sketch of proof

The operation of this criterion can be thought of as seen from the upper and lower sums of the series to be tested. The sequence is divided into blocks of ascending length and estimated in each block against maximum and minimum. Since the sequence is assumed to be monotone decreasing and the maximum and the minimum of the first to the last sequential member of each block is identical.

The criterion now follows from the comparison test. The most common block division is according to powers of two blocks. To prove convergence, we construct the majorant by

For each index k contains the majorant members of the same value, so the majorant converges if and only if converges.

In order to demonstrate divergence, we construct the minor ante by

For each index k contains the minorant elements with the same value, so the minorant diverges iff diverges.

Example of use

One application is in the general harmonic series. Has for a fixed

The same convergence behavior as

Is obviously a geometric series with a factor. From their convergence behavior follows that for convergence, or divergence is present. Note the change of the starting value and the index of the row of.

Analog results for the more slowly converging or diverging series

For convergence, or divergence.

Swell

• Konrad Konigsberg: Analysis 1 Springer Publishers, Berlin and others, 2004, ISBN 3-540-41282-4, page 78

• Convergence criterion