1/(12)+1/(23)+1/(34)+...+1/(20122013)

来源：学生作业帮助网编辑：作业帮时间：2024/05/02 12:13:15

1/(1*2)+1/(2*3)+1/(3*4)+...+1/(2012*2013)
1/(1*2)+1/(2*3)+1/(3*4)+...+1/(2012*2013)

1/(1*2)+1/(2*3)+1/(3*4)+...+1/(2012*2013)
解
1/(1*2)+1/(2*3)+1/(3*4)+...+1/(2012*2013)
=(1-1/2)+(1/2-1/3)+(1/3-1/4)+……+(1/2012-1/2013)
=1+(1/2-1/2)+(1/3-1/3)+……+(1/2012-1/2012)-1/2013——内部抵消
=1-1/2013
=2012/2013
裂项
1/n(n+1)=1/n-1/(n+1)

1/(1*2)+1/(2*3)+1/(3*4)+...+1/(2012*2013)
=1-1/2+1/2-1/3+1/3-1/4+.....+1/2012-1/2013
=1-1/2013
=2012/2013

原式=1-1/2+1/2-1/3+1/3-1/4+...+1/2012-1/2013
=1-1/2013
=2012/2013