<s>[INST] Doubt about how to compute $\sum_{n=1}^{2019}{\frac{1+2+3+4...+n}{1^3+2^3+3^3...+2019^3}}$ In the book I am reading, I have encountered the following sum.
$$S = \sum_{n=1}^{2019}{\frac{1+2+3+4...+n}{1^3+2^3+3^3...+2019^3}}$$
From here, I factored the denominator since it does not seem to be dependent on $n,$ and I rewrote the expression as
$$\frac{1}{1^3+2^3+3^3+ \cdots +2019^3} \sum_{n=1}^{2019}{1+2+3+4+ \cdots +n}=\frac{1}{1^3+2^3+3^3 + \cdots +2019^3} \sum_{n=1}^{2019}{\sum_{k=1}^n}k$$
From here, I proceeded to compute the sum since there is  an equation for the sum of the first $n$ terms, and then, I solved the sum of this equation. Also, I applied the equation of the sum of the first $n^3$ terms. Finally, my result was wrong. My question is regarding my initial approach to the problem since everything else is very simple. I have thought about other ways to solve this, and I can not find it. The official answer is
$$\frac{2019}{2010}.$$
Give this problem a try, and if you arrive at the solution, I invite you to help me understand it.
 [/INST] Let's replace $2019$ by $T$.  In the numerator:
$$ \sum_{n=1}^T \sum_{k=1}^n k = \sum_{n=1}^T \frac{n(n+1)}{2}  = \frac{(T+1)^3 - (T+1)}{6}$$
In the denominator:
$$ \sum_{k=1}^T k^3 = \frac{(T+1)^4 - 2 (T+1)^3 + (T+1)^2}{4} $$
So:
$$ \frac{ \sum_{n=1}^T \sum_{k=1}^n k }{\sum_{k=1}^T k^3} = \frac{4}{6} \frac{(T+1)^3 - (T+1)}{(T+1)^4 - 2 (T+1)^3 + (T+1)^2} = \frac{2 (T+2)}{3 T (T+1)}$$
For $T=2019$ this is $\dfrac{2021}{3030 \cdot 2019}$.  The official answer is wrong.
Or maybe the question was supposed to be
$$ \frac{1}{2}\sum_{n=1}^{2019} \frac{1 + 2 + \ldots + n}{1^3 + 2^3 + \ldots + n^3} $$
which would work out to $\dfrac{2019}{2020}$.
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