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

[lamayale]

1. Prove that

lim_n→∞ n^1/n
2. Hint: Define α_n=n^^1/n-1 and use Binomial Formula to show that for each index n, n=(1+α_n)ⁿ ≥1+(n(n-1)/2)α²_n
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 α_n=n^1/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+α_n)ⁿ) ≥∑1+(n(n-1)/2)α²_n

Is this right?


Please help me to understand the problem, it is driving me nuts. Thank you very much.

Homework Statement

 

[vanhees71]

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

The binomial theorem reads
$$(a+b)^n=\sum_{k=0}^{n} \binom{n}{k} a^k b^{n-k},$$
where the binomial coefficients are defined as
$$\binom{n}{k}=\frac{n!}{k!(n-k)!}.$$

 

[lamayale]

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

The binomial theorem reads
$$(a+b)^n=\sum_{k=0}^{n} \binom{n}{k} a^k b^{n-k},$$
where the binomial coefficients are defined as
$$\binom{n}{k}=\frac{n!}{k!(n-k)!}.$$

Yes, sorry, that was a typo. But still, I do not know the relationship. Let me fix it.