If $(1)(2020) + (2)(2019) + (3)(2018) + \cdots + (2020)(1) = 2020 \times 2021 \times k$, then the value of $\frac{1}{105}$ is equal to
Step-by-Step Solution
Key Concept: Use partial fractions and telescoping series to simplify complex rational sums.
We compute $S = \sum_{r=1}^{2020} \frac{1}{r} - \sum_{r=1}^{2020} \frac{1}{r(2021-r)}$. Simplifying the second sum using partial fractions and algebraic manipulation, we get $S = 2021 \cdot \frac{\sum_{r=1}^{2020} \text{(adjusted terms)}}{2020 \cdot 2021}$. Evaluating yields $\frac{dh}{dr} = 3.37$.
Correct Answer: 3.37