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Investigation into infinite limits

  1. Mar 23, 2009 #1
    I was looking into infinite limits and sequences and looked at the Limit
    Lim {f(n+1)/f(n)}
    n->∞
    I was looking to see if there were any significant patterns and i couldn't even solve. It was proposed to me by a friend am I just having a bad case of limit block or is it as difficult as it seems??

    p.s f(n+1) = f sub (n+1)
    f(n) = f sub n
     
  2. jcsd
  3. Mar 23, 2009 #2
    If I'm reading your notation correctly, how difficult it is depends on the form of f(n).
     
  4. Mar 23, 2009 #3
    it can be what you like and by difficult i mean mentally challenging not solvability haha
     
  5. Mar 24, 2009 #4

    HallsofIvy

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    Then what in the world is your question?
     
  6. Mar 24, 2009 #5
    surely the nature of the limit is defined by the nature of f(n)?
     
  7. Mar 24, 2009 #6
    Let f(n) = 1 for all n. Then the limit is 1.

    Let f(n) = c^n. Then the limit is 1/c.

    Let f(n) = p(x). Then the limit is 1.

    Let f(n) = sin(n). Then the limit is undefined.

    Let f(n) = (-1)^n. Then the limit is -1.

    etc.

    These were all pretty easy. I guess the answer to your question, then, may be that they're, in general, not very hard at all. Of course, I could say

    f(n) = (n^n)sin(ln(n))/(n!)(ln n!)

    Give that one a shot and let me know how it turns out.
     
  8. Mar 24, 2009 #7
    since in the infinite series n must be greater than or equal to 1 we can take the example that n=10 thus we say that
    lim f(subscript)n/f(subscript)n+1
    n->10

    is equal to the series
    (f1, f2, f3, f4, f5, f6 ,f7, f8, f9,f10)/(f1,f2,f3,f4,f5,f6,f7,f8,f9,f10.f11)

    with the common parts of each series eliminated what is left is
    1/(f11)
    thus if we say that n=x where both n and x are integers greater than or equal to 1
    so that
    lim f(subscript)n/f(subscript)n+1 = 1/(f[x+1])
    n->x
    that way we can set x=∞
    lim f(subscript)n/f(subscript)n+1 = 1/(f[∞+1]) = 1/∞ = 0
    n-> ∞
    there fore
    Lim {f(n+1)/f(n)} = 0
    n->∞

    Q.E.D??
     
  9. Mar 24, 2009 #8
    Ummm, I think you have something of a misunderstanding of what sequence, series, etc. mean and what lim(f_n / f_n+1) means.

    A sequence is a function from the (positive...let's say) integers to the real numbers.
    f: N -> R

    A sequence can have a limit as n goes to infinity if f(n) gets arbitrarily close to some finite value as you make n arbitrarily large.

    A series is a limit of partial sums of a sequence f(n) as the number of terms in the partial sums goes to infinity.

    As a counterexample to:
    Lim {f(n+1)/f(n)} = 0
    n->∞

    Try f(n) = n. Then f(n+1)/f(n) = n+1/n = 1+1/n and the limit of this is 1, not zero.
     
  10. Mar 25, 2009 #9
    hmm k i think i might have misunderstood the initial equation but you explanation makes sense thank you kindly
     
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