Limit and error term question

1. Feb 21, 2007

akoska

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

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.

2. Relevant equations

na

3. 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.

2. Feb 23, 2007

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