MathNet 2013 Question 56

Let $k \ge 2$ be an integer and $0 \lt a_1 \lt a_2 \lt \dots \lt a_k$ be real numbers. Compute $\lim_{n \to \infty} \{ \sqrt[n]{a_1 n + a_2^n + \dots + a_k^n} \}$, where $\{x\}$ denotes the fractional part of $x$.