Proving limit of the nth root of n

  • Thread starter Elzair
  • Start date
  • #1
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Homework Statement



Prove the following limit:

[tex]lim_{n \rightarrow \infty } n^{ 1 / n } = 1 [/tex]

Homework Equations



Not sure.

The Attempt at a Solution



Given any [tex]\epsilon > 0[/tex], choose [tex]N \in \mdseries N[/tex] s.t.

[tex]\left| n^{ 1 / n } - 1 \right| < \epsilon[/tex] for all [tex]n > N[/tex]

I am not sure how to proceed.
 

Answers and Replies

  • #2
jgens
Gold Member
1,583
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If you're still working on this problem and need to do it with epsilons and deltas, I think that choosing N = exp(log(1+ε)-1) should suffice. I can't find a nice/elegant epsilon delta solution to this problem, but maybe someone else can.
 
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  • #3
125
2
Prove [tex]\lim _{n\to \infty} \sqrt[n]{n}=1[/tex].
Proof: We want:
[tex]
|\sqrt[n]{n}-1|<\epsilon
[/tex]
The abs sign can be safely dropped, it follows that
[tex]
n<(1+\epsilon)^n
[/tex]
Using binomial theorem to expand the first 3 terms of RHS.
[tex]
n<1+n\epsilon+\frac{1}{2}n(n-1)\epsilon^2+...
[/tex]
As long as we make n<0.5n(n-1)εε, the first inequality holds. It requires
[tex]
n>1+\frac{2}{\epsilon^2}
[/tex]

With all that said,
For any ε>0, there exists N=[1+2/(εε)], such that if n>N, then
[tex]
|\sqrt[n]{n}-1|<\epsilon
[/tex]

Q.E.D
P.S. I love ε-δ proof:)
 
Last edited:
  • #4
micromass
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