Cho hàm số \(f\left( x \right)=\ln 2020-\ln \left( \frac{x+1}{x} \right).\) Tính \(f'\left( 1 \right)+f'\left( 2 \right)+...+f'\left( 2020 \right).\)

* Hướng dẫn giải

Ta có \(f'\left( x \right)=-\frac{x}{x+1}.\left( \frac{x+1}{x} \right)'=-\frac{x}{x+1}.\left( \frac{-1}{{{x}^{2}}} \right)=\frac{1}{\left( x+1 \right)x}=\frac{1}{x}-\frac{1}{x+1}.\)

Khi đó

\(f'\left( 1 \right)+f'\left( 2 \right)+...+f'\left( 2019 \right)+f'\left( 2020 \right)=1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+...+\frac{1}{2019}-\frac{1}{2020}+\frac{1}{2020}-\frac{1}{2021}\)

\(=1-\frac{1}{2021}=\frac{2020}{2021}.\)