Cauchy condensation test

A convergence test stating that a positive nonincreasing series converges exactly when the series of its terms at powers of two, weighted by those powers, converges.

Cauchy condensation theorem: A theorem that equates convergence of a positive nonincreasing series with convergence of its dyadically condensed series. This is the theorem’s standard name and formal statement.
p-series: A series of terms 1/n^p, whose convergence depends on whether the exponent p exceeds one. Condensation reduces its terms to a geometric series with ratio determined by p.
Infinite series: The sum of an infinite sequence of terms, defined through the limit of its partial sums. The condensation test classifies convergence of this object.
Ratio test: A convergence test based on the limit or upper limit of ratios of successive absolute terms. It can handle positive terms without requiring monotonicity, unlike condensation.
Dyadic interval: An interval whose length is a power of two and whose endpoints lie on a power-of-two grid. Blocks from 2^k through 2^(k+1)−1 underpin the comparison.
Bertrand series: A family of series with terms involving powers of n and iterated logarithms. Condensation exposes threshold exponents that are difficult to see from the original terms.
Partial sum: The sum of the first finitely many terms of a sequence or series. Convergence is defined by whether these sums approach a finite limit.
Root test: A convergence test based on the limit or upper limit of nth roots of absolute terms. It detects exponential-scale decay directly rather than grouping terms by powers of two.
Integral test: A test comparing a positive decreasing series with the improper integral of a corresponding function. Both tests exploit monotonicity to compare a series with grouped or continuous area.
Logarithmic integral: The function defined by integrating 1/log(x) from a fixed positive lower bound. Its related reciprocal-logarithm series is a standard condensation application.

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