# Categorizing the behaviour of x^n/n

• August 14th 2011, 03:03 PM
Melsi
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:

$\frac{(1.4)^{2}}{2}=0.97$

$\frac{(1.4)^{3}}{3}=0.91 (smaller)$

$\frac{(1.4)^{4}}{4}=0.96 (bigger)$

$\frac{(1.4)^{5}}{5}=1.07$

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!
• August 14th 2011, 03:27 PM
Plato
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 $\frac{x^n}{n}$ converge?
• August 14th 2011, 06:06 PM
Also sprach Zarathustra
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:

$\frac{(1.4)^{2}}{2}=0.97$

$\frac{(1.4)^{3}}{3}=0.91 (smaller)$

$\frac{(1.4)^{4}}{4}=0.96 (bigger)$

$\frac{(1.4)^{5}}{5}=1.07$

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!

If the question you asking is similar to Plato question, then suppose that x>0 and prove:

i) If $0 then $\lim_{n\to\infty}\frac{x^n}{n}=0$.

ii) If $x>1$, then $\lim_{n\to\infty}\frac{x^n}{n}=\infty$.

But, I suspect that you unfamiliar with the limit definition.
• August 15th 2011, 02:03 AM
Melsi
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!