Limit regarding $\sum\limits_{k=1}^n ka_k$

Apply Stolz–Cesàro theorem ! Let $a_n = \sum \limits_{k=1}^n k a_k$ and $b_n = n^2$ .

The limit in question, $\lim\limits_{n\to\infty} \frac{a_n}{b_n}$ exists, if and only if the limit below exist. And if so, they are equal to each other.

$\lim\limits_{n\to\infty} \frac{a_{n+1} - a_n}{b_{n+1} - b_n}$

We have $\lim\limits_{n\to\infty} \frac{a_{n+1} - a_n}{b_{n+1} - b_n} =\lim\limits_{n\to\infty} \dfrac{(n+1)a_n}{2n + 1} =\lim\limits_{n\to\infty} \dfrac{(1 + 1/n) \cdot a_n}{2 + 1/n} = \dfrac{ (1 + 0) \cdot 2020}{2 + 0} = \boxed{1010}.$