Basic Analysis I — Real Analysis · Problem 2.6.12

Sequences and Series — Jiri Lebl

Mathematics — Real Analysis · alternating-series

Problem

(Challenging): Show that the monotonicity in the alternating series test is necessary. That is, find a sequence of positive real numbers

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

with

\lim_{n \to \infty} x_n = 0

but such that

\sum_{n=1}^{\infty} (-1)^n x_n

diverges.