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Proof the lim n^(1/n) using a hint

  1. Jan 21, 2014 #1
    1. Prove that

    limn→∞ n1/n



    2. Hint: Define αn=n^1/n-1 and use Binomial Formula to show that for each index n, n=(1+αn)n ≥1+(n(n-1)/2)α2n



    3. It is clear to me that the limit is 1, and I thought that if I prove that 1/n goes to 0, then I can just say the limit of n goes to 1 and keep going with my life, but I am supposed to use the hint that is confusing me.

    My questions are mostly regarding the hint:

    1) It perfectly makes sense to try lim αn=n1/n-1 to make it zero. I am fine with that, and I perfectly understand that.

    2) I am unable to understand why the hint is there and how it is related to the problem at all.
    Assigning n=1,2,3 (I did it with excel) to the hint, I got that for each term the inequality is true. I still do not know at all why I should use it or how it is related with this problem.

    3) They say to use the binomial formula, which I guess they mean on the hint and to do something like this:

    (1+αn)n) ≥∑1+(n(n-1)/2)α2n

    Is this right?


    Please help me to understand the problem, it is driving me nuts. Thank you very much.
    1. The problem statement, all variables and given/known data
     
    Last edited: Jan 21, 2014
  2. jcsd
  3. Jan 21, 2014 #2

    vanhees71

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    2016 Award

    I don't understand, why you put a sum symbol in your item 3).

    The binomial theorem reads
    [tex](a+b)^n=\sum_{k=0}^{n} \binom{n}{k} a^k b^{n-k},[/tex]
    where the binomial coefficients are defined as
    [tex]\binom{n}{k}=\frac{n!}{k!(n-k)!}.[/tex]
     
  4. Jan 21, 2014 #3
    Yes, sorry, that was a typo. But still, I do not know the relationship. Let me fix it.
     
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