# Homework Help: Show lim((2n)&^(1/n)) = 1

1. Mar 12, 2012

### Hodgey8806

1. The problem statement, all variables and given/known data
I need to show the lim((2n)^(1/n)) = 1

2. Relevant equations
I will be using the definition of the limit as well as using the Binomial Theorem as an aide.

I am following an example from my book quite similar. So applying the Binomial Theorem to this problem, I will choose to write (2n)^(1/n) as 1 + Kn for some Kn > 0.

Raising both sides to the n power, we have 2n = (1 + Kn)^n ≥ 1 + (1/2)n(n-1)(Kn)^2
=> 2n ≥ (1/2)n(n-1)(Kn)^2
I will then solve for Kn to get that Kn≤ 2/√n-1

This tells us that there is some Nε such that 2/√Nε-1 < ε since ε>0 (By the Archimedean Property).

3. The attempt at a solution
Now, applying that to my proof, I have:
Let ε>0 be given.
0<(2n)^(1/n) -1 = (1 + Kn) -1 = Kn ≤ 2/√n-1 < ε
Since ε>0 is arbitrary, we can conclude that lim((2n)^(1/n)) = 1

I appreciate the help! I would like to know if this is mostly correct, and would like help in rewriting it to make it neater. Thank you :)