Question 3
Let $N = 2012$. Find all $N^2$-uples of real numbers $a_{i,j}$, $1 \le i, j \le N$, such that the following limit is convergent:
$$
\lim_{x \to +\infty} \sum_{1 \le i,j \le N} \left(a_{i,j} \sqrt[j]{x+i}\right)
$$
$$
\lim_{x \to +\infty} \sum_{1 \le i,j \le N} \left(a_{i,j} \sqrt[j]{x+i}\right)
$$