Lebl-Basic-Analysis-I · Problem 2.5.6

Sequences and Series

Mathematics · ratio-test

Problem

Prove the following stronger version of the ratio test: Let ∑n=1∞xn\sum_{n=1}^{\infty} x_n be a series. a) If there is an NN and a ρ<1\rho < 1 such that ∣xn+1∣∣xn∣<ρ\frac{|x_{n+1}|}{|x_n|} < \rho for all n≥Nn \geq N, then the series converges absolutely. (Remark: Equivalently the condition can be stated as lim sup⁡n→∞∣xn+1∣∣xn∣<1\limsup_{n\to\infty} \frac{|x_{n+1}|}{|x_n|} < 1.) b) If there is an NN such that ∣xn+1∣∣xn∣≥1\frac{|x_{n+1}|}{|x_n|} \geq 1 for all n≥Nn \geq N, then the series diverges.

Related theory

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