Categorizing the behaviour of x^n/n
Hello,
I've spent many hours today investigating a sequence, I finally reduced it to (x^n)/n which is the smallest it gets, for obvious reasons.
My observations for x=1.4:
^{2}}{2}=0.97)
^{3}}{3}=0.91 (smaller))
^{4}}{4}=0.96 (bigger))
^{5}}{5}=1.07 )
I checked on pc an I believe it keeps being strictly increasing for the rest of the terms.
My questions:
1. Is this a wide know behaviour, if so, any reference to this category for studying further would be a great help.
2. How do I call this sequence, increasing or decreasing because it gets strictly increasing after some point.
3. I suspect it does not converge because the terms are aproaching infinity, am I right?
Thank you very much!
Re: Categorizing the behaviour of x^n/n
Quote:
Originally Posted by
Melsi 
I've spent many hours today investigating a sequence, I finally reduced it to (x^n)/n which is the smallest it gets, for obvious reasons.
This is not being mean. This is just an observation.
I don’t see a question in all of that. What is the question?
Unless you post an exact question, we cannot be expected to help.
Is the question this: For what values of x does 
converge?
Re: Categorizing the behaviour of x^n/n
Quote:
Originally Posted by
Melsi Hello,
I've spent many hours today investigating a sequence, I finally reduced it to
(x^n)/n which is the smallest it gets, for obvious reasons.
My observations for x=1.4: I checked on pc an I believe it keeps being strictly increasing for the rest of the terms. My questions:
1. Is this a wide know behaviour,
if so, any reference to this category for studying further would be a great help.
2. How do I call this sequence, increasing or decreasing because it gets strictly increasing after some point.
3. I suspect it does not converge because the terms are aproaching infinity, am I right?
Thank you very much!
If the question you asking is similar to Plato question, then suppose that x>0 and prove:
i) If 
then 
.
ii) If 
, then 
.
But, I suspect that you unfamiliar with the limit definition.
Re: Categorizing the behaviour of x^n/n
Sorry guys you are right, my mistake!
It was when I wanted to check the behaviour of x^n/n on the pc and I was getting strange results(for example it started as decreasing and then became an increasing sequence).
It was this the very fact that convinced me I was going to need some help, forgetting that one should not apply things that are true for convergings sequences only.
Yes a useful question could be for what values it converges, that is answeared very clearly already! Thank you Zaratoustra!
My sincere apologies again,
Thank you guys!
