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?

  • $\begingroup$ You mention a $P > 1$ but it is not present anywhere in the series. From the wording, I would expect the sum of $x_n^P$ $\endgroup$ – Will Jagy Dec 19 '18 at 22:30
  • $\begingroup$ Sorry.I edited it now $\endgroup$ – DD90 Dec 19 '18 at 22:31

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

  • 2
    $\begingroup$ Thank you! so afterwards we can use the convergence of the p- series and the comparison test! $\endgroup$ – DD90 Dec 19 '18 at 22:39

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.