\(\frac{1}{x\left(x+1\right)}+\frac{1}{\left(x+1\right)\left(x+2\right)}+\frac{1}{\left(x+2\right)\left(x+3\right)}+...+\frac{1}{\left(x+2019\right)\left(x+2020\right)}\) 

( ĐKXĐ : \(x\ne\left\{0;-1;-2;...;-2019;-2020\right\}\))

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

\(=\frac{1}{x}-\frac{1}{x+2020}\)

\(=\frac{x+2020}{x\left(x+2020\right)}-\frac{x}{x\left(x+2020\right)}\)

\(=\frac{x+2020-x}{x\left(x+2020\right)}\)

\(=\frac{2020}{x\left(x+2020\right)}\)

                                                           Bài giải

\(\frac{1}{x\left(x+1\right)}+\frac{1}{\left(x+1\right)\left(x+2\right)}+...+\frac{1}{\left(x+2019\right)\left(x+2020\right)}\)

\(=\frac{1}{x}-\frac{1}{x+1}+\frac{1}{x+1}-\frac{1}{x+2}+...+\frac{1}{x+2019}-\frac{1}{x+2020}\)

\(=\frac{1}{x}-\frac{1}{x+2020}\)

\(=\frac{x+2020}{x\left(x+2020\right)}-\frac{x}{x+2020}=\frac{2020}{x\left(x+2020\right)}\)

\(\frac{1}{x\left(x+1\right)}+\frac{1}{\left(x+1\right)\left(x+2\right)}+\frac{1}{\left(x+2\right)\left(x+3\right)}+...+\frac{1}{\left(x+2019\right)\left(x+2020\right)}\)

\(=\frac{1}{x}-\frac{1}{x+1}+\frac{1}{x+1}-\frac{1}{x+2}+\frac{1}{x+2}-\frac{1}{x+3}+...+\frac{1}{x+2019}-\frac{1}{x+2020}\)

\(=\frac{1}{x}-\frac{1}{x+2020}=\frac{x+2020}{x\left(x+2020\right)}-\frac{x}{x\left(x+2020\right)}=\frac{2020}{x\left(x+2020\right)}\)

ủa bạn j ơi chữ x chành bành ra trên đề kìa mà bạn bảo tìm làm j nữa

Tham khảo tại: Câu hỏi của Lương Đức Hưng - Toán lớp 8 | Học trực tuyến

Đặt \(a=\frac{1}{1^2}+\frac{1}{2^2}+...+\frac{1}{2019^2}\)

\(b=\frac{1}{2^2}+\frac{1}{3^2}+...+\frac{1}{2020^2}\)

Khi đó : \(D=ab-\left(b+1\right)\left(a-1\right)\)

\(\Rightarrow D=ab-\left(ab+a-b-1\right)\)

\(\Rightarrow D=b-a+1=\frac{1}{2020^2}-1+1=\frac{1}{2020^2}\)

Xét \(f\left(x\right)+f\left(1-x\right)=\frac{x^3}{1-3x+3x^2}+\frac{\left(1-x\right)^3}{1-3\left(1-x\right)+3\left(1-x\right)^2}\)

\(=\frac{x^3}{1-3x+3x^2}+\frac{1-3x+3x^2-x^3}{1-3+3x+3-6x+3x^2}\)

\(=\frac{x^3}{1-3x+3x^2}+\frac{1-3x+3x^2-x^3}{1-3x+3x^2}\)

\(=\frac{1-3x+3x^2}{1-3x+3x^2}=1\)

Thay vào ta tính được:

\(A=\left[f\left(\frac{1}{2020}\right)+f\left(\frac{2019}{2020}\right)\right]+...+\left[f\left(\frac{1009}{2020}\right)+f\left(\frac{1011}{2020}\right)\right]+f\left(\frac{1010}{2020}\right)\)

\(A=1+...+1+f\left(\frac{1010}{2020}\right)\) (với 1009 số 1)

\(A=1009+f\left(\frac{1}{2}\right)=1009+\frac{\left(\frac{1}{2}\right)^3}{1-3\cdot\frac{1}{2}+3\cdot\left(\frac{1}{2}\right)^2}\)

\(A=1009+\frac{1}{2}=\frac{2019}{2}\)

Vậy \(A=\frac{2019}{2}\)

Tks bạn nhé

hello ae xin chào