Cauchy Condensation Test
Compress slowly converging positive terms into dyadic blocks with honest bounds.
By Riano · Published · Updated · Report a correction
Cauchy Condensation Test
Concept Overview
The Cauchy Condensation Test is a standard convergence test in calculus, allowing us to determine the convergence of a series with positive, monotonically decreasing terms. The test compresses the sequence into "blocks" whose sizes are powers of 2. For a sequence an, it asserts that the sum of an converges if and only if the sum of 2ka2k converges.
Worked Example
Consider the series defined by an = 1 / (n * (ln n)p) for n ≥ 2. We use blocks n = 2k to 2k+1 - 1. The upper bound of the k-th block is 2k * a2k = 1 / (k ln 2)p. Let us calculate for p = 1 and K = 1 block (k=1, which covers n=2 to 3):
Original terms for block 1 (n=2, 3): a_2 = 1 / (2 * (ln 2)^1) ≈ 0.72134752 a_3 = 1 / (3 * (ln 3)^1) ≈ 0.30341307 Partial sum = 0.72134752 + 0.30341307 = 1.02476059 Upper bound: c_1 = 1 / (1 * ln 2)^1 ≈ 1.44269504 Lower bound: c_2 / 2 = (1 / (2 * ln 2)^1) / 2 ≈ 0.36067376
Experiment
Open the Cauchy Condensation Test module.
- Set the Log exponent p control to 1.
- Set the Dyadic blocks control to 4.
- Observe that the Original partial sum is approximately 2.033.
- Notice the Infinite series result is Divergent, since p = 1.
- Change the Log exponent p to 2. The series now reports Convergent.
Limitations
Finite partial sums never prove convergence alone; monotonicity and positivity justify the condensation. The test is strictly applicable only to non-increasing sequences of non-negative terms.
References
Primary sources for the Cauchy Condensation Test and real analysis foundations.
Experience it interactively
Adjust parameters, observe in real time, and build deep intuition with Riano’s interactive Cauchy Condensation Test module.
Try Cauchy Condensation Test on Riano →