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Math Help - Categorizing the behaviour of x^n/n

  1. #1
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    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!
    Last edited by Melsi; August 14th 2011 at 03:06 PM. Reason: typographical errors
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  2. #2
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    Re: Categorizing the behaviour of x^n/n

    Quote Originally Posted by Melsi View Post
    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 donít 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?
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  3. #3
    MHF Contributor Also sprach Zarathustra's Avatar
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    Re: Categorizing the behaviour of x^n/n

    Quote Originally Posted by Melsi View Post
    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<x<1 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.
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  4. #4
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    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!
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