Theorem: (Cauchy Condensation Test) If is a monotonically decreasing sequence of positive reals and is a positive integer, then
converges if and only if converges.
Problem: Determine the convergence of , where is a positive real.
Solution: Well, let’s apply the Cauchy condensation test. Then we know that
converges if and only if
does. But this clearly diverges due to the fact that the numerator is exponential and the denominator is a power function. QED.
Comment: This is a pretty powerful test for convergence, at least in the situations in which it can be applied. The non-calculus proof for the divergence of the harmonic series is very similar to the Cauchy condensation test; in fact, the condensation test would state that
converges iff converges,
which clearly shows that the harmonic series diverges.
Practice Problem: Determine the convergence of .