Investigation into infinite limits

In summary, the author was looking into infinite limits and sequences and looked at the Limit Lim {f(n+1)/f(n)} n->∞and found that it is not always very difficult to solve the limit. Of course, the difficulty may depend on the form of the limit.
  • #1
Drake13
5
0
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
 
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  • #2
If I'm reading your notation correctly, how difficult it is depends on the form of f(n).
 
  • #3
it can be what you like and by difficult i mean mentally challenging not solvability haha
 
  • #4
Then what in the world is your question?
 
  • #5
surely the nature of the limit is defined by the nature of f(n)?
 
  • #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.
 
  • #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??
 
  • #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.
 
  • #9
hmm k i think i might have misunderstood the initial equation but you explanation makes sense thank you kindly
 

1. What is an infinite limit?

An infinite limit is a mathematical concept that describes the behavior of a function as the input approaches a certain value (usually infinity). It indicates that the function either approaches a positive or negative infinity, or oscillates between these values.

2. How do you investigate infinite limits?

To investigate infinite limits, you can use various mathematical techniques such as evaluating the limit algebraically, graphically, or numerically. You can also use properties of limits, such as the squeeze theorem, to determine the behavior of a function as the input approaches infinity.

3. What is the importance of studying infinite limits?

Studying infinite limits is important in mathematics because it helps us understand the behavior of functions and their limits as the input approaches infinity. This knowledge is crucial in many areas of mathematics, such as calculus and differential equations.

4. Can infinite limits be used in real-life applications?

Yes, infinite limits can be used in real-life applications. For example, they can help us understand the behavior of physical phenomena, such as the velocity of an object as it approaches the speed of light. They can also be used in economics and finance to model the growth of investments or populations.

5. Are there any limitations to investigating infinite limits?

While infinite limits are a useful concept in mathematics, there are some limitations to their investigation. One limitation is that not all functions have well-defined infinite limits, so they may not behave as expected. Additionally, infinite limits do not always provide an accurate representation of real-world phenomena, as they are based on idealized mathematical models.

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