Basic Analysis I — Real Analysis · Problem 2.5.15

Sequences and Series — Jiri Lebl

Mathematics — Real Analysis · cauchy-condensation

Problem

(Challenging): Suppose

\{x_n\}_{n=1}^{\infty}

is a decreasing sequence of positive numbers. The proof of convergence/divergence for the

p

-series generalizes. Prove the so-called Cauchy condensation principle:

\sum_{n=1}^{\infty} x_n \quad \text{converges if and only if} \quad \sum_{n=1}^{\infty} 2^n x_{2^n} \quad \text{converges.}