设直线kx+(k+1)y=2√505(k为正整数)与两坐标轴围成的图形面积为Sk,

问题描述:

设直线kx+(k+1)y=2√505(k为正整数)与两坐标轴围成的图形面积为Sk,
S1+S2+S3+…+S2019=?

Sk=1/2 (2√505 / k) (2√505 / (k+1) )=1010/( k(k+1) )S1+S2+S3+…+S2019=1010( 1/1·1/2+1/2·1/3+…+1/2019·1/2020 )=1010( (1/1-1/2)+(1/2-1/3)+…+(1/2019-1/2020) )=1010(1-1/2020)=2019/2