Hi I am having trouble proving the following inequality:

Let n be a natural number.

$\displaystyle n^{\frac{1}{n}} + n^{\frac{1}{(n+1)}} + ... + n^{\frac{1}{(2n - 1)}} \geq n^{\sqrt[n]{2}}$

The book I found this problem in suggests the use of these mean values:

$\displaystyle A(x_1,x_2,...,x_n) = \frac{x_1+x_2+...+x_n}{n}$

and

$\displaystyle G(x_1,x_2,...,x_n) = \sqrt[n]{x_1x_2...x_n}$

I have tried everything but cant get it to work!

Any suggestions?