Basic Analysis I — Real Analysis · Problem 2.5.13

Sequences and Series — Jiri Lebl

Mathematics — Real Analysis · series-partial-sums

Problem

Let sks_k be the kkth partial sum of ∑n=1∞xn\sum_{n=1}^{\infty} x_n. a) Suppose there exists an m∈Nm \in \mathbb{N} such that lim⁡k→∞smk\lim_{k \to \infty} s_{mk} exists and lim⁡n→∞xn=0\lim_{n \to \infty} x_n = 0. Show that ∑n=1∞xn\sum_{n=1}^{\infty} x_n converges. b) Find an example where lim⁡k→∞s2k\lim_{k \to \infty} s_{2k} exists and lim⁡n→∞xn≠0\lim_{n \to \infty} x_n \neq 0 (and therefore ∑n=1∞xn\sum_{n=1}^{\infty} x_n diverges). c) (Challenging) Find an example where lim⁡n→∞xn=0\lim_{n \to \infty} x_n = 0, and there exists a subsequence {ski}i=1∞\{s_{k_i}\}_{i=1}^{\infty} such that lim⁡i→∞ski\lim_{i \to \infty} s_{k_i} exists, but ∑n=1∞xn\sum_{n=1}^{\infty} x_n still diverges.

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