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: Convergence of a series

  1. Jan 30, 2010 #1
    Hey guys,
    1. The problem statement, all variables and given/known data
    Prove: If for every n [tex] a_{n}>0 [/tex] and [tex] \frac{a_{n+1}}{a_{n}}<1 [/tex] then the series [tex]lim_{n->\infty} a_{n}<0 [/tex]

    3. The attempt at a solution
    We know that [tex] a_{n} [/tex] is lowerly bounded by 0 and upwardly bounded by [tex] a_{1} [/tex]. we also know that it is monotonic and decreasing and so congerges. But how do I show that it converges to 0. What is to stop it from converging to, say, .5?
    Last edited: Jan 30, 2010
  2. jcsd
  3. Jan 30, 2010 #2
    When you say series, are you referring to a summation or a sequence?

    If an < 0 for all n, then all your terms are negative. So apply this fact to an+1/an < 1
  4. Jan 30, 2010 #3
    Sorry, I made a typo. that was an>0.
  5. Jan 30, 2010 #4
    and I am refering to a sequence, not a summation. Pardon me, english is not my native language.
  6. Jan 30, 2010 #5
    Ahh, I misread the question. it was prove or disprove. I found a counter example.
    Thanks anyway.
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook