极数计算（要过程）

来源：百度知道 编辑：UC知道 时间：2024/05/21 21:00:30
1/1*3+1/3*5+1/5*7+.........+1/2003*2005=?
1/2+1/2+4+1/2+4+6+.........+1/2+4+6+...+200=?
1/2+1/(2+4)+1/(2+4+6)+.........+1/(2+4+6+...+200)=?

1/1*3+1/3*5+1/5*7+.........+1/2003*2005

=1/2(1-1/3)+1/2(1/3-1/5)+1/2(1/5-1/7)+.....+1/2(1/2003-2005)

=1/2(1-1/3+1/3-1/5+1/5-1/7+.....+1/2003-1/2005)

=1/2(1-1/2005)

=1/2*2004/2005

=1002/2005

1/2+1/(2+4)+1/(2+4+6)+.........+1/(2+4+6+...+200)

∵2+4+6+.....+2n=(2+2n)n/2=n(n+1)

∴1/(2+4+....+2n)=1/n(n+1)=1/n-1/(n+1)

∴原式=1/1*2+1/2*3+1/3*4+....+1/100*101

=1-1/2+1/2-1/3+1/3-1/4+...+1/100-1/101

=1-1/101

=100/101