Lebl-Basic-Analysis-I · Problem 2.5.13

Sequences and Series

Mathematics · 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.
Intuitive explanation
PART (A)
The key insight is that if every mm-th partial sum converges and the terms vanish, you can "fill in the gaps" between these subsequential limits. Since xn→0x_n \to 0, the partial sums between smks_{mk} and sm(k+1)s_{m(k+1)} stay close together, so the full sequence of partial sums must also converge by the Cauchy criterion.
PART (B)
You need terms that oscillate in a way that makes every other partial sum stable, yet prevent the series from converging. Think about a sequence where consecutive pairs of terms cancel or nearly cancel—like alternating signs with non-vanishing magnitude—so s2ks_{2k} stabilizes while s2k−1s_{2k-1} drifts.
PART (C)
The challenge here is to construct terms that go to zero but arrange them so that only a sparse subsequence of partial sums converges, while the full sequence diverges. Consider how you might have long "blocks" of small terms that accumulate slowly, with a carefully chosen subsequence that happens to land at points where the cumulative sum stabilizes.
Intuitive explanation
PART (A)
The key insight is that if every mm-th partial sum converges and the terms vanish, you can "fill in the gaps" between these subsequential limits. Since xn→0x_n \to 0, the partial sums between smks_{mk} and sm(k+1)s_{m(k+1)} stay close together, so the full sequence of partial sums must also converge by the Cauchy criterion.
PART (B)
You need terms that oscillate in a way that makes every other partial sum stable, yet prevent the series from converging. Think about a sequence where consecutive pairs of terms cancel or nearly cancel—like alternating signs with non-vanishing magnitude—so s2ks_{2k} stabilizes while s2k−1s_{2k-1} drifts.
PART (C)
The challenge here is to construct terms that go to zero but arrange them so that only a sparse subsequence of partial sums converges, while the full sequence diverges. Consider how you might have long "blocks" of small terms that accumulate slowly, with a carefully chosen subsequence that happens to land at points where the cumulative sum stabilizes.

Related theory

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