My specific problem:

For fixed $x\in(0,1),$ I would like to know how large $n$ has to be in terms of $x$ so that $n(1-x)^{n-2}\leq \tfrac{1}{5}.$ Since $\displaystyle\lim\_{n\rightarrow\infty}n(1-x)^{n-2}=0$ this must eventually be true. 

I think that this can be accomplished using some upper bound on $n(1-x)^n$ but I haven't found any that are tight enough. For example, the bound $(1-x)^n\leq \frac{1}{1+nx}$ won't help here. Thanks in advance for your help!