1. ## convergent/divergent series

$\displaystyle \sum_{k=0}^{\infty} \sqrt[n]2 -1$

i'm supposed to test this series for convergence or divergence

i just learned the root test (i'm assuming maybe i should use it here?) and i've reviewed the examples in my book but i'm not really sure how to apply it.. help please?

2. Originally Posted by buttonbear
$\displaystyle \sum_{k=0}^{\infty} \sqrt[n]2 -1$

i'm supposed to test this series for convergence or divergence

i just learned the root test (i'm assuming maybe i should use it here?) and i've reviewed the examples in my book but i'm not really sure how to apply it.. help please?
Your looking at comparing the series (limit comparison test)
$\displaystyle \sum_{n=1}^\infty 2^{\frac{1}{n}}-1$ with $\displaystyle \sum_{n=1}^\infty \frac{1}{n}$

3. using the limit comparison test, don't you get that the limit of an/bn as n->infinity is zero? and, if that's correct, where do you go from there? i thought the limit comparison test was only conclusive for c>0

4. Originally Posted by buttonbear
using the limit comparison test, don't you get that the limit of an/bn as n->infinity is zero? and, if that's correct, where do you go from there? i thought the limit comparison test was only conclusive for c>0
$\displaystyle \lim_{n \to \infty} \frac{2^\frac{1}{n}-1}{\frac{1}{n}}$ and if we let $\displaystyle n = \frac{1}{x}$ then $\displaystyle \lim_{x \to 0} \frac{2^x-1}{x} = \ln 2$ and so by the limit comparison test, the original series diverges.