设直线kx+(k+1)y-1=0(k是正整数)与坐标围成的三角形面积为Sk,则S1+S2+S3……S2013的值是?

kx+(k+1)y-1=0
x/(1/k)+y/(1/(k+1))=1
所以
sk=1/2 *1/k(k+1)
从而
s1+s2+.+s2013
=1/2 (1/1×2＋1/2×3＋.+1/2013×2014)
=1/2 (1-1/2+1/2-1/3+.+1/2013-1/2014)
=1/2 (1-1/2014)
=1/2 ·2013/2014
=2013/4028