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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?

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

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    $\begingroup$ Thank you! so afterwards we can use the convergence of the p- series and the comparison test! $\endgroup$
    – DD90
    Dec 19, 2018 at 22:39
  • $\begingroup$ @DD90 not "afterwards". the convergence of the p-series and comparison test is used in the "it suffices to show". $\endgroup$
    – mathworker21
    Aug 17, 2019 at 10:11

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