Investigation into infinite limits

[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

 

[csprof2000]

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

 

[Drake13]

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

 

[HallsofIvy]

Then what in the world is your question?

 

[2^Oscar]

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

 

[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.

 

[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??

 

[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.

 

[Drake13]

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