1. Not finding help here? Sign up for a free 30min 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!

I can't seem to understand the ratio test proof

  1. Apr 3, 2015 #1
    Hi everyone, I'm currently taking Calc II course and I'm kind of stuck in this ratio test proof thing.
    1. The problem statement, all variables and given/known data

    http://blogs.ubc.ca/infiniteseriesmodule/appendices/proof-of-the-ratio-test/proof-of-the-ratio-test/

    I'm trying to understand the proof, but there are some parts that I don't really get.
    So assuming that |an+1/an| = L < 1
    and that L < r
    |an+1/an| < r
    but then, the proof says there is an integer N which is < or = to n. From where did that come from? Why do we need it in the first place? And how does it relate to the proof?
    The proof goes on and I don't really have any idea what's going on.
    2. Relevant equations
    -
    3. The attempt at a solution
    I tried making a number line to understand what's going on but this seems so subtle.
     
  2. jcsd
  3. Apr 3, 2015 #2
    there are some small typos in the proof which maybe make it harder to understand it, but anyway that N comes from the definition of the limit of a sequence (this is stated in the proof that it follows from the formal definition of the limit). Intuitively when we know that a sequence Cn has limit L this means that Cn gets closer to L as n grows larger. So if we want Cn to be close enough to L so that it is L<Cn<r all we have to do is to choose a large enough N and for all n>=N it will be L<Cn<r.

    The proof utilizes of this fact for the sequence Cn=|an+1/an|. It uses this N to prove that |aN+k|<|aN|rk. From this last inequality you can prove that the series [itex]\sum_{k=1}^{\infty}|a_{N+k}|[/itex] converges (because it is bounded by the geometric series [itex]\sum_{k=1}^{\infty}|a_{N}|r^k[/itex] which obviously converges). Now the series [itex]\sum_{k=1}^{\infty}|a_{k}|[/itex] also converges because it differs from the [itex]\sum_{k=1}^{\infty}|a_{N+k}|[/itex] only by a finite sum [itex]\sum_{k=1}^{N}|a_k|[/itex]
     
    Last edited: Apr 3, 2015
  4. Apr 3, 2015 #3

    vela

    User Avatar
    Staff Emeritus
    Science Advisor
    Homework Helper
    Education Advisor

    The proof doesn't say ##\lvert a_{n+1}/a_n \rvert = L##. It says that
    $$\lim_{n \to \infty} \left\lvert \frac{a_{n+1}}{a_n} \right\rvert = L.$$ If you understand what the limit means, the proof should make more sense to you.
     
  5. Apr 3, 2015 #4
    Thanks, that makes much more sense now!

    Vela: yea sorry, I knew about the limit I just forgot to type it.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted



Similar Discussions: I can't seem to understand the ratio test proof
Loading...