Basic Analysis I — Real Analysis · Problem 2.5.9

Sequences and Series — Jiri Lebl

Mathematics — Real Analysis · absolute-convergence

Problem

a) Prove that if ∑n=1∞xn\sum_{n=1}^{\infty} x_n and ∑n=1∞yn\sum_{n=1}^{\infty} y_n converge absolutely, then ∑n=1∞xnyn\sum_{n=1}^{\infty} x_n y_n converges absolutely. b) Find an explicit example where the converse does not hold. c) Find an explicit example where all three series are absolutely convergent, are not just finite sums, and (∑n=1∞xn)(∑n=1∞yn)≠∑n=1∞xnyn\left(\sum_{n=1}^{\infty} x_n\right)\left(\sum_{n=1}^{\infty} y_n\right) \neq \sum_{n=1}^{\infty} x_n y_n. That is, show that series are not multiplied term-by-term.

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