Limit and error term question

[akoska]

Homework Statement

Compute the limit a_n=n^(1/n) without using the fact that lim log a_n=log lim a_n. Instead, we're expected to solve this using binomial theorem and the error term.

Homework Equations

na

The Attempt at a Solution

Well, the error term = |a_n - L| which is expected to go to zero. I tried using binomial theorem on ((n-1)+1)^1/n with no luck. Also tried squeeze theorem.

 

[gammamcc]

hint

The binomial theorem is not obvious since you do now have whole number exponents.

Here is an attempt:
a_n=n^{1/n}

a_{n+1}-a_n = (n+1)^{1/(n+1)}-n^{1/n}

<(n+1)^{1/n}-n^{1/n}


and a_{n+1}-a_n > (n+1)^{1/(n+1)}-(n)^{1/(n+1)}
Showing that these comparitive terms go to 0 as n goes to infinity may use the "Binomial Number" expansion. Then, Sandwich Thm applies.

http://en.wikipedia.org/wiki/Binomial_theorem