Lebl-Basic-Analysis-I · Problem 2.3.15

Sequences and Series

Mathematics · limit-superior-limit-inferior-bolzano-weierstrass

Problem

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 lim sup⁡n→∞∣xn+1∣∣xn∣<1,\limsup_{n\to\infty} \frac{|x_{n+1}|}{|x_n|} < 1, then {xn}n=1∞\{x_n\}_{n=1}^\infty converges to 0. b) Prove that if lim inf⁡n→∞∣xn+1∣∣xn∣>1,\liminf_{n\to\infty} \frac{|x_{n+1}|}{|x_n|} > 1, then {xn}n=1∞\{x_n\}_{n=1}^\infty is unbounded.

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