Proving the Limit of 2^(1/n) = 1

[tarheelborn]

Homework Statement

How do I prove that lim 2^(1/n) = 1?

Homework Equations

The Attempt at a Solution

 

[foxjwill]

What have you done so far?

 

[tarheelborn]

This is actually part of a series problem, so I have determined that the sequence of a_n's is decreasing and that it seems to converge to 1. I know that I need to find N \in the naturals such that |2^(1/n)| < epsilon, but I can't seem to figure out how to solve in terms of epsilon.

 

[foxjwill]

Since $2^{1/n}$ is always greater than 1 (prove this!), you can forget the absolute value. Try taking the natural log of both sides of $2^{1/n}-1<\epsilon$ and solving for $n$.