(Cauchy Condensation Test) Let (ak​) be a monotonically decreas... | Filo

SOLVED:(Cauchy Condensation Test) Let (ak) be a monotonically decreasing sequence of nonnegative real numbers. Show that the series ∑k ≥1 ak is convergent if and only if the series ∑k ≥0 2^k a2^k is convergent. (Hint: Proposition 9.4.) Deduce the convergence and divergence of the series ∑k ≥1 1 / k^p and ∑k ≥2 1 / k(lnk)^p, where p ∈ℝ. (Compare Example 9.43.)