Investigation into infinite limits

1. Mar 23, 2009

Drake13

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. Mar 23, 2009

csprof2000

If I'm reading your notation correctly, how difficult it is depends on the form of f(n).

3. Mar 23, 2009

Drake13

it can be what you like and by difficult i mean mentally challenging not solvability haha

4. Mar 24, 2009

HallsofIvy

Staff Emeritus
Then what in the world is your question?

5. Mar 24, 2009

2^Oscar

surely the nature of the limit is defined by the nature of f(n)?

6. Mar 24, 2009

csprof2000

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.

7. Mar 24, 2009

Drake13

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??

8. Mar 24, 2009

csprof2000

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.

9. Mar 25, 2009

Drake13

hmm k i think i might have misunderstood the initial equation but you explanation makes sense thank you kindly