Cauchy condensation test
For decreasing : converges if and only if converges.
Proposition 1.36 (Cauchy Condensation Test [Non-Examinable])
Let for all , and suppose that is decreasing. Then
Proof
The substitution gives , so by the integral test converges exactly when exists.
Let . Since is decreasing, for ,
and integrating over and summing,
Hence the integrals converge exactly when converges, and the result follows.
Related
Stated in
- Proposition 1.36 (Cauchy Condensation Test [Non-Examinable])§1.4 Series and Convergence Tests