TITLE: How prove this limit $\displaystyle\lim_{n\to \infty}a_{n}=1$
QUESTION [10 upvotes]: Let $a_n$ be the only positive root of the equation $x^n+x=1$, for each $n\in \Bbb N$.
Show that $\lim \limits_{n\to \infty}a_{n}$ exists,and find its value.
My guess is that
$$\lim \limits_{n\to \infty}a_{n}=1$$
But How prove that it exists, and how find it?

REPLY [5 votes]: Technically, one should first verify that $a_n$ is indeed unique.
This can be done easiest by Descartes' Rule of Signs, a less-known but in my opinion underrated result which is used as follows.  Note that the polynomial $x^n + x - 1$ has coefficients $1, 1, -1$ in that order (ignore $0$ coefficients), so the coefficients signs go $+,+, -$ (positive, positive, negative).  The coefficients change sign exactly once, so the number of positive real roots of the polynomial $x^n + x - 1$ is one.  (In general, if the coefficients change sign $n$ times, the number of positive real roots will be one of $n$, $n - 2$, $n - 4$, $n - 6$, etc.)
Once you have established that $a_n$ is unique, Rebecca's solution completes the proof.