In the book Applied Econometric Time Series by Walter Enders (third edition, page 30) there is a discussion about the characteristic polynomial of the homogeneous part of an n-th order difference (recurrence) equation. In the discussion, one necessary and one sufficient condition for roots of the polynomial
$$\alpha^{n} – a_{1}\alpha^{n-1} – a_{2}\alpha^{n-2}… – a_{n}$$
to lie inside the unit circle (roots are allowed to be complex, so roots with magnitude less than 1) are presented.
Put differently, we seek the values of $\alpha$ that solve
$$\alpha^{n} – a_{1}\alpha^{n-1} – a_{2}\alpha^{n-2}… – a_{n} = 0,$$
and the claims are as follows:
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In an n-th order equation, a necessary condition for all characteristic roots to lie inside the unit circle is $$\sum\limits_{i = 1}^{n} a_{i} < 1.$$
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Since the values of the $a_{i}$ can be positive or negative, a sufficient condition for all characteristic roots to lie inside the unit circle is $$\sum\limits_{i = 1}^{n} |a_{i}| < 1.$$
I would very much appreciate references or explanations that prove the claims. I have tried to see if I could show the claims directly by symbol manipulations and using the assumptions in the claims, but I have not tried using any general theorems. I am not very knowledgeable about polynomials in general (which might be obvious from this post), so I realize that I might not possess the necessary mathematical machinery to understand these claims. Thank you in advance to anyone that took the time to read this post.
Edit: The answer from Hagen von Eitzen below helped me understand (2) and partly (1). However, I still haven't understood (1) completely. Perhaps I have used his hint differently than intended. My confusion is outlined in a comment below his reply.
Best Answer
For 1.: Intermediate value theorm on $[1,\infty)$
For 2.: triangle inequality for $|\alpha|=|a_1+\alpha^{-1}a_2+\cdots|$.