# Homework Help: Prove n^(1/n) tends to 1 as n tends to infinity

1. Nov 21, 2008

### heshbon

1. The problem statement, all variables and given/known data

Need to prove n^(1/n) tend to 1 as n tends to infinty

2. Relevant equations

3. The attempt at a solution

Last edited: Nov 21, 2008
2. Nov 21, 2008

### HallsofIvy

The sequence $n^{1/n}$, as n goes to infinity, converges to a if the function $x^{1/x}$ converges to a as x goes to infinity. If we set $y= x^{1/x}$ then ln(y)= (ln x)/x which is of the "infinity/infinity" form so we can use L'Hopital's rule.

3. Nov 21, 2008

### Pere Callahan

One direct method that comes to my mind is to show that for any $\varepsilon>0$ there exist N such that
$$n^{1/n}\leq 1+\varepsilon$$
(It is easy to see that
$$n^{1/n}\geq 1$$
)
for all n>N.
The first equation is equivalent to $n\leq(1+\varepsilon)^n=1+n\varepsilon+\dots$

Do you see how to choose N?

4. Nov 21, 2008

### heshbon

wow..briliantly simple using l'hopital...though i have not yet come across this theorem at uni...still will impress the tutors. thanks

5. Nov 21, 2008

### heshbon

I cant see how to choose N....could you give me another hint?

6. Nov 21, 2008

### Pere Callahan

I should have included the next term in the binomial expansion

$$n\leq 1+n\varepsilon+\frac{n(n-1)}{2}\varepsilon^2\Leftrightarrow \dots$$
You just have to solve this for n>... and take the next larger integer for N.