I am thinking about the following questions:
$A$ is a complex $n\times n$ matrix, $\rho(A)$ means the spectral radius of $A$, $\operatorname{tr}(A)$ means the trace of $A$, then prove: $$ \varlimsup_{k\to\infty} \big( |\operatorname{tr}(A^k)|^{1/k} \big) = \rho(A). $$
It is clearly equivalent to $$ \varlimsup_{k\to\infty} \left| \sum_{i=1}^n x_i^k \right|^{1/k} = \max_{1\le i\le n} |x_i|, \quad x_i\in\mathbb{C}, i=1,\cdots,n. $$ When every $x_i\in\mathbb{R}$, it is easy. But when all of $x_i$ are complex, I can only obtain $$ \varlimsup_{k\to\infty} \left| \sum_{i=1}^n x_i^k \right|^{1/k} \le \max_{1\le i\le n} |x_i|. $$ Any hint?