设直线kx+（k+1）y=1（k是正整数）与两坐标轴所围城图形的面积为S1,S2,……,S2010,则S1+S2+……+S2010

Question

设直线kx+（k+1）y=1（k是正整数）与两坐标轴所围城图形的面积为S1,S2,……,S2010,则S1+S2+……+S2010

Answer 1

直线y=kx+k-1与x轴交点：((1-k)/k,0)
直线y=（k+1）x+k与x轴交点：(-k/(k+1),0)
直线y=kx+k-1和直线y=（k+1）x+k交点：(-1,-1)
Sk=1/[2k(k+1)]
S1+S2+...+S2010
=1/(2•1•2)+1/(2•2•3)+...+1/(2•2010•2011)
=(1/2)(1-1/2+1/2+1/3+...+1/2010-1/2011)
=1005/2011