1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: Sequence satisfying a condition for all n

  1. Sep 22, 2009 #1
    1. The problem statement, all variables and given/known data

    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.

    2. Relevant equations

    3. The attempt at a solution
  2. jcsd
  3. Sep 22, 2009 #2


    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    You can start with s2 > as1. Now what about s3? Can you compare it to s2 and s1? Continue...
  4. Sep 22, 2009 #3


    User Avatar
    Science Advisor

    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"?
  5. Sep 22, 2009 #4
    If [tex](s_n)[/tex] is a sequence and the limit [tex]\lim_{n \to \infty}|s_{n+1} / {s_n}| = L [/tex] exists and [tex]L < 1[/tex], then [tex]\lim s_n[/tex] converges. If not, what do you think happens?
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook