1. PF Contest - Win "Conquering the Physics GRE" book! Click Here to Enter
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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


    User Avatar
    Science Advisor
    Gold Member
    2017 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
  4. Jan 21, 2014 #3
    Yes, sorry, that was a typo. But still, I do not know the relationship. Let me fix it.
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted