Cauchy condensation test explained

Cauchy condensation test should not be confused with Cauchy's convergence test.

f(n) of non-negative real numbers, the series $\backslash sum\backslash limits\_^\; f(n)$ converges if and only if the "condensed" series $\backslash sum\backslash limits\_^\; 2^\; f(2^)$ converges. Moreover, if they converge, the sum of the condensed series is no more than twice as large as the sum of the original.

Estimate

The Cauchy condensation test follows from the stronger estimate,$$\backslash sum\_^\; f(n)\; \backslash leq\; \backslash sum\_^\; 2^n\; f(2^n)\; \backslash leq\backslash \; 2\backslash sum\_^\; f(n),$$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. That term is always the first one, since by assumption the terms are non-increasing.

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 $2\; \backslash sum\_^\; f(n)$ which begins with $f(2^)$ lines up with the end of the run of $\backslash sum\_^\; 2^\; f(2^)$ which ends with $f(2^)$, so that the former stays always "ahead" of the latter.

Integral comparison

The "condensation" transformation $f(n)\; \backslash rarr\; 2^\; f(2^)$ recalls the integral variable substitution $x\; \backslash rarr\; e^$ yielding $f(x)\backslash ,\backslash mathrmx\; \backslash rarr\; e^\; f(e^)\backslash ,\backslash mathrmx$.

f , that $\backslash sum\backslash limits\_^f(n)$ converges if and only if \displaystyle\int ^{inf ty} _{1} f(x) d x converges. The substitution $x\backslash rarr\; 2^x$ yields the integral \displaystyle log 2 \int ^{inf ty} _{2} 2 ^{x} f( 2 ^{x} ) d x . We then notice that \displaystyle log 2 \int ^{inf ty} _{2} 2 ^{x} f( 2 ^{x} ) d x < log 2 \int ^{inf ty} _{0} 2 ^{x} f( 2 ^{x} ) d x , where the right hand side comes from applying the integral test to the condensed series $\backslash sum\backslash limits\_^\; 2^f(2^)$. Therefore, $\backslash sum\backslash limits\_^\; f(n)$ converges if and only if $\backslash sum\backslash limits\_^\; 2^f(2^)$ converges.

Examples

The test can be useful for series where appears as in a denominator in . For the most basic example of this sort, the harmonic series $\backslash sum\_^\; 1/n$ is transformed into the series $\backslash sum\; 1$, which clearly diverges.

As a more complex example, take$$f(n)\; :=\; n^\; (\backslash log\; n)^\; (\backslash log\; \backslash log\; n)^.$$

Here the series definitely converges for, and diverges for . When, the condensation transformation gives the series$$\backslash sum\; n^\; (\backslash log\; n)^.$$

The logarithms "shift to the left". So when, we have convergence for, divergence for . When the value of enters.

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

k = 1 , 2 , 3 ,\ldots , the generalized Bertrand series $$\backslash sum\_\; \backslash frac\; \backslash quad\backslash quad\; (N=\backslash lfloor\; \backslash exp^\; (0)\; \backslash rfloor+1)$$ converges for \alpha > 1 and diverges for 0 < \alpha \leq 1 .^[1] Here f ^{\circ} denotes the th iterate of a function f , so that The lower limit of the sum, N , 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. For instance, in the case of k = 2 and \alpha = 1 , the partial sum exceeds 10 only after 10 ^{10 ^{100}} (a googolplex) terms; yet the series diverges nevertheless.

Schlömilch's generalization

A generalization of the condensation test was given by Oskar Schlömilch.^[2] Let be a strictly increasing sequence of positive integers such that the ratio of successive differences is bounded: there is a positive real number, for which

Then, provided that

f(n) meets the same preconditions as in Cauchy's convergence test, the convergence of the series $\backslash sum\_^\; f(n)$ is equivalent to the convergence of$$\backslash sum\_^\; \backslash ,\; f(u(n))\; \backslash \; =\backslash \; \backslash sum\_^\; \backslash Big(u(n1)-u(n)\backslash Big)\; f(u(n)).$$

Taking $u(n)\; =\; 2^n$ so that $\backslash Delta\; u(n)\; =\; u(n1)-u(n)\; =\; 2^n$, the Cauchy condensation test emerges as a special case.

References

• Bonar, Khoury (2006). Real Infinite Series. Mathematical Association of America. .

External links

• Cauchy condensation test proof

Notes and References

1. Book: Rudin, Walter . Principles of Mathematical Analysis . McGraw-Hill . 1976 . 0-07-054235-X . New York . 62–63.
2. Elijah Liflyand, Sergey Tikhonov, & Maria Zeltse (2012) Extending tests for convergence of number series page 7/28 via Brandeis University