# Thread: Proof of limit of nth root of n

1. ## Proof of limit of nth root of n

My lecturer has asked us to prove:

im confused as to what details I need to provide as it looks as if its pretty much complete bar the end? Have I missed something?

2. ## Re: Proof of limit of nth root of n

Originally Posted by carla1985
My lecturer has asked us to prove:

im confused as to what details I need to provide as it looks as if its pretty much complete bar the end? Have I missed something?

This is the standard proof. Can you do i, ii, & iii ?
Note that if $n\ge 2$ then
$0 \leqslant \left( {\sqrt[n]{n} - 1} \right)^2 \leqslant \frac{4}{n}$.

So, $\left( {\frac{4}{n}} \right) \to 0\;\therefore \,\left( {\sqrt[n]{n} - 1} \right)^2 \to0\text{ or }\left({\sqrt[n]{n}} \right) \to 1$

3. ## Re: Proof of limit of nth root of n

we have barely looked at binomial theorem yet so I'm not completely sure what he's asking of me. unfortunately my calculus lecturer isnt brilliant at making the questions follow the class work. I can follow what is going on:
i) the series is just expanded out
ii) n/2 is substituted in for n-1 and rearranged
iii) x is substituted.

i think thats it?

4. ## Re: Proof of limit of nth root of n

Originally Posted by carla1985
we have barely looked at binomial theorem yet so I'm not completely sure what he's asking of me. unfortunately my calculus lecturer isnt brilliant at making the questions follow the class work. I can follow what is going on:
i) the series is just expanded out
ii) n/2 is substituted in for n-1 and rearranged
iii) x is substituted.

So your difficulty begins with (i).
$\left( {1 + x} \right)^n = \sum\limits_{k = 0}^n {\binom{n}{k}x^k }.$

Any term is the sum is less than or equal to $(1+x)^n$
so $(1+x)^n\ge\binom{n}{2}x^2=\frac{n(n-1)}{2}x^2$.

5. ## Re: Proof of limit of nth root of n

ok, i understand that

6. ## Re: Proof of limit of nth root of n

Originally Posted by carla1985
ok, i understand that

Then what more do you want?
Are you waiting for someone like Soroban or johng to come along and give you a camera ready complete proof? Well wait on. I don't do that. I think it defeats the purpose of learning from doing.

7. ## Re: Proof of limit of nth root of n

not at all, you've misunderstood me. I said I understand as I thought I understood your post, so went back to work some more on the question. I apologise if I have managed to offend you but I was simply replying to be polite while carrying on trying to get my head around the whole thing. I'm not asking for a whole answer, I've been out of education for over 10 years and and struggling to keep up so came to the forum with help in understanding. Also the homeworks are a very small part of the course as the majority of it is exam based so just having answers just handed to me isnt going to pass my exams either, I'm no teenager that can just throw away their degree or get someone else to give them the work n scrape through, this means a lot to me so if I'm posting a question its because I genuinly am struggling to either understand what the question is asking (as in this case) or what I need to do to get the answer. And even if given an answer I never just copy it down, I go away and try to understand it too.

Aside from that, thank you very much for your help, I am very grateful x

8. ## Re: Proof of limit of nth root of n

Another proof :

ln(n^(1/n))=lnn/n therefore lim(lnn/n)=0 (Hospital's ule) and consequently lim(n)^(1/n)=1

MINOAS

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