1*3*5分之1+3*5*7分之1+5*7*9分之1+7*9*11分之1+9*11*13分之1+11*13*15分之1如何简算?

一般地,1/[(2n-1)(2n+1)(2n+3)]=(1/3){1/[(2n-1)(2n+1)]-1/[(2n+1)(2n+3)]},
令n=1,2,3,4,5,6,相加即得
原式=(1/3){[1/(1×3)-1/(3×5)]+[1/(3×5)-1/(5×7)]+[1/(5×7)-1/(7×9)]+[1/(7×9)-1/(9×11)]
+[1/(9×11)-1/(11×13)]+[1/(11×13)-1/(13×15)]}
=(1/3)[1/(1×3)-1/(13×15)]=64/585.