The discussion revolves around a sequence of positive numbers {s_n} that satisfies the condition s_(n+1) > αs_n for all n, where α > 1. Participants are tasked with demonstrating that s_n approaches infinity as n increases.
The discussion is ongoing, with participants sharing their thoughts on how to approach the problem. Some guidance has been offered regarding comparisons between terms, but there is no explicit consensus on the best method to proceed.
Participants are working under the constraints of a homework assignment, which may limit the information they can use or the methods they can apply. There is also a mention of a limit condition related to convergence that is being questioned in the context of the problem.
So [itex]s_2> a s_n[/itex], [itex]s_3> a s_2> a(a s_1)= a^2 s_1[/itex], [itex]s_4> a s_3> a(a^2 s_1)= a^3 s_1[/itex]. So [itex]s_n>[/itex] a to what power times [itex]s_1[/itex]? What does that have to do with a "geometric sequence"?StarTiger said:Homework Statement
Suppose that a sequence {s_n} of positive numbers satisfies the condition s_(n+1) > αs_n for all n where α > 1. Show that s_n → ∞
My teacher mentioned something about making it into a geometric sequence and taking the log. I'm just confused.
Homework Equations
The Attempt at a Solution