# Comparing geometric progression with arithmetic progression to determine common ratio

Background: I have been studying a proof offered by Driscoll (1999) that shows that if $$x'Ax \sim \chi^2(rank(A), \frac12 \mu'A\mu)$$ then $$AV$$ is idempotent of rank $$r$$. Within this proof (necessity), he points out that (equation 6b on page 274):

$$\sum_{i=1}^P (\lambda_i^2 + nv_i^2)\lambda_i^{n-2} = d + n\gamma,\qquad n = 2,3,...$$

In this case, $$\lambda_i$$ represents the eigenvalues $$i=1...p$$ of $$AV$$. Considering this equation only for even values of $$n$$, Driscoll then argues if $$|\lambda_i| > 1$$, for any $$i$$, then the left side grows geometrically in $$n$$ whereas the right side grows arithmetically. He therefore concludes that $$-1 \leqslant \lambda_i \leqslant 1$$ for all $$i$$.

Question: Following this argument, can we really conclude that $$|\lambda_i|$$ must be equal or smaller than $$1$$? I would argue that if we compare the nature of growth on both sides then we can only conclude that $$|\lambda_i| =1$$.

Of course, if $$|\lambda_i| > 1$$ the left side grows exponentially. On the other hand, however, if $$|\lambda_i| < 1$$ then the left side should converge, shouldn't it? And in this case, growth on both sides is again 'not aligned' for $$n$$ increasing. In fact, it is only aligned (i.e., arithmetic) if $$|\lambda_i| = 1$$. Is there anything I misunderstand?