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