Proving an inequality involving exponentiation

Homework Statement

Show that $$\left( 1 - \frac{\ln n}{kn} \right)^n > \frac{1}{n^{1/k} + 1}$$ holds for all integers $$n\geq 1$$ and $$k\geq 2$$.

The Attempt at a Solution

I first tried to find a proof for $$k=2$$ by showing that the quotient LHS/RHS goes to 1 and has negative slope everywhere, but this becomes rather unwieldy.