I'm stuck on this problem. Any advice is appreciated:

i. Prove that $\forall n \in \mathbb{N}$ the equation $x^n +x - 1 = 0$ has a unique solution $r_n$.

ii. Find the limit of the sequence $(r_n)_{n\geq1}$.

Here's what I have so far (not much, sadly)

I rewrote the equation in i. to get 1 to the other side. I also believe that the solution should be positive for the few numbers that I have tried. I'm thinking about using Newton's Method but I'm not sure how effective it would be. Any advice?

  • $\begingroup$ (i) isn't even true. For $n = 2$, and it appears for all even $n$, there are two distinct solutions. $\endgroup$ – Mike Pierce Mar 15 '16 at 2:44
  • $\begingroup$ And indeed, when $n$ is even there are always two solutions. $\endgroup$ – Robert Israel Mar 15 '16 at 2:48
  • 1
    $\begingroup$ Wait, I'm sorry. I missed something. It should say "unique positive." my apologies. $\endgroup$ – Rethink123 Mar 15 '16 at 2:49

Hint for (1): increasing function.

Hint for (2): Is $r_n > 1$ or $< 1$? For fixed $x$ in the appropriate interval, what happens to $x^n + x - 1$ as $n \to \infty$ what does that tell you about how $r_n$ relates to $x$?

  • $\begingroup$ For (1), would using Newton's Method work? I was considering this because of the fact that it's increasing and that the second derivative is positive (except for n=1, I think). $\endgroup$ – Rethink123 Mar 15 '16 at 2:53
  • $\begingroup$ No, you don't need to find the solution, you need to know that it exists and is unique. This is much more basic. $\endgroup$ – Robert Israel Mar 15 '16 at 2:55
  • $\begingroup$ Would Intermediate Value Theorem work to show that a positive solution exists and then a proof by contradiction to show that it's unique work? $\endgroup$ – Rethink123 Mar 15 '16 at 3:01
  • $\begingroup$ Yes, that's a good plan. $\endgroup$ – Robert Israel Mar 15 '16 at 7:13
  • $\begingroup$ I'm having trouble with the second part. I know that the sequence is bounded between 0 and 1. I also know that it's increasing, but I'm not sure how to prove it. I also think that the limit is 1, but I'm not sure how to prove this part either. $\endgroup$ – Rethink123 Mar 15 '16 at 17:28

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