Cauchy condensation test

In mathematics, the Cauchy condensation test, named after Augustin-Louis Cauchy, is a standard convergence test for infinite series.

of non-negative real numbers, the series

converges if and only if the "condensed" series

Moreover, if they converge, the sum of the condensed series is no more than twice as large as the sum of the original.

The Cauchy condensation test follows from the stronger estimate,

which should be understood as an inequality of extended real numbers.

The essential thrust of a proof follows, patterned after Oresme's proof of the divergence of the harmonic series.

To see the first inequality, the terms of the original series are rebracketed into runs whose lengths are powers of two, and then each run is bounded above by replacing each term by the largest term in that run.

To see the second inequality, these two series are again rebracketed into runs of power of two length, but "offset" as shown below, so that the run of

lines up with the end of the run of

recalls the integral variable substitution

Pursuing this idea, the integral test for convergence gives us, in the case of monotone

, where the right hand side comes from applying the integral test to the condensed series

The test can be useful for series where n appears as in a denominator in f. For the most basic example of this sort, the harmonic series

As a more complex example, take

Here the series definitely converges for a > 1, and diverges for a < 1.

When a = 1, the condensation transformation gives the series

The logarithms "shift to the left".

This result readily generalizes: the condensation test, applied repeatedly, can be used to show that for

, the generalized Bertrand series

denotes the mth iterate of a function

The lower limit of the sum,

, was chosen so that all terms of the series are positive.

Notably, these series provide examples of infinite sums that converge or diverge arbitrarily slowly.

, the partial sum exceeds 10 only after

(a googolplex) terms; yet the series diverges nevertheless.

A generalization of the condensation test was given by Oskar Schlömilch.

[2] Let u(n) be a strictly increasing sequence of positive integers such that the ratio of successive differences is bounded: there is a positive real number N, for which

meets the same preconditions as in Cauchy's convergence test, the convergence of the series

, the Cauchy condensation test emerges as a special case.

Visualization of the above argument. Partial sums of the series ${\textstyle \sum f(n)}$ , ${\textstyle \sum 2^{n}f(2^{n})}$ , and ${\textstyle 2\sum f(n)}$ are shown overlaid from left to right.