Lebl-Basic-Analysis-I · Problem 2.2.13

Sequences and Series

Mathematics · facts-about-limits-of-sequences

Problem

(Easy) Prove the following stronger version of Lemma 2.2.12, the ratio test. Suppose {xn}n=1∞\{x_n\}_{n=1}^\infty is a sequence such that xn≠0x_n \neq 0 for all nn. a) Prove that if there exists an r<1r < 1 and M∈NM \in \mathbb{N} such that ∣xn+1∣∣xn∣≤rfor all n≥M,\frac{|x_{n+1}|}{|x_n|} \leq r \quad \text{for all } n \geq M, then {xn}n=1∞\{x_n\}_{n=1}^\infty converges to 0. b) Prove that if there exists an r>1r > 1 and M∈NM \in \mathbb{N} such that ∣xn+1∣∣xn∣≥rfor all n≥M,\frac{|x_{n+1}|}{|x_n|} \geq r \quad \text{for all } n \geq M, then {xn}n=1∞\{x_n\}_{n=1}^\infty is unbounded.

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