1/2-1/6-1/12-1/20-1/30-1/42-···-1/132

1/2-1/6-1/12-1/20-1/30-1/42-···-1/132
1/2-1/6-1/12-1/20-1/30-1/42-···-1/132

1/2-1/6-1/12-1/20-1/30-1/42-···-1/132
1/2-1/6-1/12-1/20-1/30-1/42-···-1/132
=（1-1/2）-（1/2-1/3）-（1/3-1/4）-（1/4-1/5）-（1/5-1/6）-（1/6-1/7）-...-（1/11-1/12）
=（1-1/2-1/2）+1/3-1/3+1/4-1/4+1/5-1/5+1/6-1/6+1/7-.-1/11+1/12
=0+1/12
=1/12

=1/2 -(1/2-1/3)-(1/3-1/4)-...-(1/11-1/12)
=1/12

1/2-1/6-1/12-1/20-1/30-1/42-···-1/132
=1/2-（1/2-1/3）-(1/3-1/4)-(1/4-1/5)-(1/5-1/6)-···-（1/11-1/12）
=1/2-1/2+1/3-1/3+1/4-1/4+1/5-1/5+1/6-···-1/11+1/12
=1/12

观察数列可知
1/6 = 1/（2*3）= 1/2 - 1/3
1/12 = 1/（3*4）=1/3 - 1/4
......
1/132 = 1/（11*12）=1/11 - 1/12
所以 原式 = 1/2 - 1/2 + 1/3 - 1/3 + … - 1/11 + 1/12
= 1/12