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Ratio Test for Series Proof

  1. May 31, 2008 #1
    1. The problem statement, all variables and given/known data
    Show that if [tex] Lim|\frac{a_{n+1}}{a_{n}}| = L > 1, [/tex] then [tex]{a_{n}\rightarrow \infty[/tex] as [tex]n\rightarrow\infty [/tex]

    Also, from that, deduce that [tex]a_{n}[/tex] does not approach 0 as [tex]n \rightarrow \infty [/tex].

    2. Relevant equations
    The book suggests showing some number r>1 such that for some number N, [tex]|a_{n+1}|> r|a_{n}|[/tex] for all n >N.
  2. jcsd
  3. May 31, 2008 #2


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    Okay, and what have you done on this problem? Have you shown, perhaps by induction on n, that [itex]|a_{n+1}|> r|a_n|[/itex]? Once you've done that, you might consider the "comparison test".
  4. May 31, 2008 #3
    How would I start that proof by induction? How can I verify that [itex]
    |a_{2}|> r|a_{1}|
    [/itex]. Also, for the second part, once I show that [tex]|a_{n}|[/tex] tends to [tex]\infty[/tex] isn't it basic logic that [tex]a_{n}[/tex] cannot approach 0?
  5. Jun 1, 2008 #4


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    You can't. It's not necessarily true. However, since the limit [itex]a_{n+1}/a_n[/itex] is less than 1, it must be true for some N. Start your induction from that.

    Yes, it is. Just state the basic logic.
  6. Jun 1, 2008 #5
    Do you mean greater than 1, or am I really missing something? And how would I start the induction? Just that for some N, [itex]|a_{2}|>r|a_{1}|[/itex]?
  7. Jun 1, 2008 #6
    If L>1, then is the sequence [tex]{a_{n}}[/tex] bounded or unbounded?
    Suppose not, If L<1, then what happens when [tex]{\lim }\limits_{n \to \infty } a_{n}[/tex]?
  8. Jun 1, 2008 #7
    If L>1 then the sequence would be unbounded right? Because the next larger term is always of a greater magnitude than the previous. If L is less than 1, then the sequence is bounded, and the limit goes to 0?
  9. Jun 1, 2008 #8

    Now, since book suggested: show that [tex] |a_{n+1}|> r|a_{n}|[/tex] for all indices [tex]n\geq N[/tex], you can use the Binomial formula to show that the sequences is unbounded. Hope that's clear.
  10. Jun 1, 2008 #9
    i've always wondered this, but how do you guys get all those math symbols in there? like the absolute value symbol, or the greater than equal to sign?
  11. Jun 1, 2008 #10
  12. Jun 16, 2008 #11
    What do you mean by the Binomial Formula? I'm still kind of confused after taking several days away.
  13. Jun 17, 2008 #12
    Well, you can use Bernoulli's Inequality, which is [tex](1+b)^n \geq 1 +nb[/tex]

    Suppose that [tex] L>1[/tex], then define [tex]b=\frac{L+1}{2}[/tex] since b<L. There exists a natural number N such that
    [tex]\frac{a_{n+1}}{a_{n}} \geq b[/tex] for all indices [tex]n \geq N[/tex] (just a reiteration of the problem)

    From here, use the Bernoulli's inequality to show that for some k and let r = [tex]b^k[/tex], then [tex]|a_{N+k}| \geq r|a_{N}|[/tex] which implies that the sequence [tex]a_{n}[/tex](the hint that your book gave) is unbounded.
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