Sequence satisfying a condition for all n

[StarTiger]

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

 

[LCKurtz]

You can start with s₂ > as₁. Now what about s₃? Can you compare it to s₂ and s₁? Continue...

 

[HallsofIvy]

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

So $s_2> a s_n$, $s_3> a s_2> a(a s_1)= a^2 s_1$, $s_4> a s_3> a(a^2 s_1)= a^3 s_1$. So $s_n>$ a to what power times $s_1$? What does that have to do with a "geometric sequence"?

 

[fmam3]

If $$(s_n)$$ is a sequence and the limit $$\lim_{n \to \infty}|s_{n+1} / {s_n}| = L$$ exists and $$L < 1$$, then $$\lim s_n$$ converges. If not, what do you think happens?