Cauchy Condensation Test
Compress slowly converging positive terms into dyadic blocks with honest bounds.
Free theory articleCompress slowly converging positive terms into dyadic blocks with honest bounds.
Concept & interpretation
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.
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.