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:**

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

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

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

$\displaystyle \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!

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 $\displaystyle \frac{x^n}{n}$ 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:**

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

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

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

$\displaystyle \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 $\displaystyle 0<x<1$ then $\displaystyle \lim_{n\to\infty}\frac{x^n}{n}=0 $.

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

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!