Convergence of Roots for an analytic function

Question

Show that the roots of $$ f(z) = z^n+z^3+z+2 =0 $$ converge to the circle $|z|=1$ as $n \to \infty$.

Answer 1

The quantity $$g(z) = z^3 + z + 2$$ has no zeros in the open unit disk and $$f_n(z) = z^n + z^3 + z + 2 \stackrel{n\to\infty}{\longrightarrow} g(z)$$ uniformly on compact subsets of the open unit disk. As a consequence we can show that, for any $0 < \epsilon \leq 1$, there is an $N\in \mathbb N$ such that, for $n \geq N$, $f_n(z)$ has no zeros in the disk $|z| \leq 1-\epsilon$. Thus the accumulation points of the zeros of the polynomials $f_n(z)$, if there are any, must lie in $|z| \geq 1$.

Similarly, $$h_n(z) = z^n f(1/z) \stackrel{n\to\infty}{\longrightarrow} 1$$ uniformly on compact subsets of the open unit disk, so we may likewise conclude that any accumulation points of the zeros of the polynomials $h_n(z)$ must lie in $|z| \geq 1$.

Combining these two observations yields the result.

Answer 2

Intuitively, if $|z| \gt 1$, taking a large power will make it very large, so much so that the other terms can't bring the sum down to zero. If $|z| \lt 1$, taking a large power will make it very small, so we can ignore it. Then we need $z=-1$ to be a root of what is left,but that violates what we already said about $z$.

To make this more rigorous, use the triangle inequality. To have a root, we need $|z|^n=|z^3+z+2|$ and so on.