1/1*2+1/2*3+1/3*4+1/4*5+1/5*6+1/6*7+1/7*8...1/49*50

1/1*2+1/2*3+1/3*4+1/4*5+1/5*6+1/6*7+1/7*8...1/49*50
1/1*2+1/2*3+1/3*4+1/4*5+1/5*6+1/6*7+1/7*8...1/49*50

1/1*2+1/2*3+1/3*4+1/4*5+1/5*6+1/6*7+1/7*8...1/49*50
1/1*2+1/2*3+1/3*4+1/4*5+1/5*6+1/6*7+1/7*8...1/49*50
=(1-1/2)+(1/2-1/3)+(1/3-1/4)+(1/4-1/5)+(1/5-1/6)+(1/6-1/7)+(1/7-1/8)+……+(1/49-1/50)
=1-1/2+1/2-1/3+1/3-1/4+1/4-1/5+1/5-1/6+1/6-1/7+1/7-1/8+……+1/49-1/50
=1-1/50
=49/50

原式=1-1/50=49/50

裂项相消法
1/1*2+1/2*3+1/3*4+1/4*5+1/5*6+1/6*7+1/7*8...1/49*50
=1-1/2+1/2-1/3+。。。+1/49-1/50
=1-1/50
=49/50