Question

1/(101*103)+1/(103*105)+1/(105*107)+.+1/(199*201)

Answer 1

1/(101*103)+1/(103*105)+1/(5*7)+1/(7*9)+……+1/(199*201)

裂项法

1/(101*103)=1/2*(1/101-1/103)

1/(103*105)=1/2*(1/103-1/105)

.

1/(199*201)=1/2(1/199-1/201)

原式

=1/2(1/101-1/103+1/103-1/105.+1/199-1/201)

=1/2(1/101-1/201)

=50/(101*201)

=50/20301