Cauchy condensation test
Encyclopedia
In mathematics
, the Cauchy condensation test, named after Augustin-Louis Cauchy, is a standard convergence test for infinite series. For a positive non-increasing sequence
f(n), the sum
converges if and only if the sum
converges. Moreover, in that case we have
A geometric view is that we are approximating the sum with trapezoid
s at every
. Another explanation is that, as with the analogy between finite sums and integral
s, the 'condensation' of terms is analogous to a substitution of an exponential function. This becomes clearer in examples such as
Here the series definitely converges for a > 1, and diverges for a < 1. When a = 1, the condensation transformation essentially gives the series
The logarithms 'shift to the left'. So when a = 1, we have convergence for b > 1, divergence for b < 1. When b = 1 the value of c enters.
. The condensation test follows from noting that if we collect the terms of the series into groups of lengths 
, each of these groups will be less than 
by monotonicity. Observe,
We have used the fact that the sequence an is non-increasing, thus
whenever 
. The convergence of the original series now follows from direct comparison to this "condensed" series. To see that convergence of the original series implies the convergence of this last series, we similarly put,
And we have convergence, again by direct comparison. And we are done. Note that we have obtained the estimate
This proof is a generalization of Oresme's proof of the divergence of the harmonic series
.
be an infinite real series whose terms are positive and non-increasing, and let 
be a strictly increasing sequence of positive integers such that
is bounded, where
is the forward difference
. Then the series
converges if the series
converges.
Taking
, we see 
, so the Cauchy condensation test emerges as a special case.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, the Cauchy condensation test, named after Augustin-Louis Cauchy, is a standard convergence test for infinite series. For a positive non-increasing sequence
Sequence
In mathematics, a sequence is an ordered list of objects . Like a set, it contains members , and the number of terms is called the length of the sequence. Unlike a set, order matters, and exactly the same elements can appear multiple times at different positions in the sequence...
f(n), the sum
converges if and only if the sum
converges. Moreover, in that case we have
A geometric view is that we are approximating the sum with trapezoid
Trapezoid
In Euclidean geometry, a convex quadrilateral with one pair of parallel sides is referred to as a trapezoid in American English and as a trapezium in English outside North America. A trapezoid with vertices ABCD is denoted...
s at every
. Another explanation is that, as with the analogy between finite sums and integralIntegral
Integration is an important concept in mathematics and, together with its inverse, differentiation, is one of the two main operations in calculus...
s, the 'condensation' of terms is analogous to a substitution of an exponential function. This becomes clearer in examples such as
Here the series definitely converges for a > 1, and diverges for a < 1. When a = 1, the condensation transformation essentially gives the series
The logarithms 'shift to the left'. So when a = 1, we have convergence for b > 1, divergence for b < 1. When b = 1 the value of c enters.
Proof
Let f(n) be a positive, non-increasing sequence of real numbers. To simplify the notation, we will write an = f(n). We are to investigate the series. The condensation test follows from noting that if we collect the terms of the series into groups of lengths , each of these groups will be less than by monotonicity. Observe,We have used the fact that the sequence an is non-increasing, thus
whenever . The convergence of the original series now follows from direct comparison to this "condensed" series. To see that convergence of the original series implies the convergence of this last series, we similarly put,And we have convergence, again by direct comparison. And we are done. Note that we have obtained the estimate
This proof is a generalization of Oresme's proof of the divergence of the harmonic series
Harmonic series (mathematics)
In mathematics, the harmonic series is the divergent infinite series:Its name derives from the concept of overtones, or harmonics in music: the wavelengths of the overtones of a vibrating string are 1/2, 1/3, 1/4, etc., of the string's fundamental wavelength...
.
Generalizations
The following generalization is due to Schlömilch. Let be an infinite real series whose terms are positive and non-increasing, and let be a strictly increasing sequence of positive integers such thatis bounded, where
is the forward differenceFinite difference
A finite difference is a mathematical expression of the form f − f. If a finite difference is divided by b − a, one gets a difference quotient...
. Then the series
converges if the seriesconverges.
Taking
, we see , so the Cauchy condensation test emerges as a special case.