# $x^n+nx-1$ has a unique solution

Show that for any integer $$n\geq1,$$ the equation $$x^n+nx-1=0$$ has a unique positive solution $$x_n$$. Furthermore, show that $$x_n$$ is such that for any $$p>1$$ the series $$\sum_{n=1}^{\infty}x_n^p$$ is convergent.

For the first part of the question I can prove the solution by the intermediate value theorem (by considering $$x=0$$ and $$x=1$$). And also uniqueness is achieved because the function is increasing (since the first derivative is always positive.)

But how about the convergence of the series?

It suffices to show $$x_n < \frac{1}{n}$$. But it is, since $$(\frac{1}{n})^n+n\frac{1}{n}-1 > 0$$.