简便计算：1155*(5/2*3*4+7/3*4*5+…+17/8*9*10+19/9*10*11)
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(2n-1)/[(n-1)n(n+1)]=1/[n(n+1)]+1/[(n-1)(n+1)]
1/[n(n+1)]=1/n-1/(n+1)
1/[(n-1)(n+1)]=1/2*[1/(n-1)-1/(n+1)]
1155*(5/2*3*4+7/3*4*5+…+17/8*9*10+19/9*10*11)
=1155*(1/3*4+1/1*3+1/4*5+1/3*5+.........+1/9*10+1/8*10+1/10*11+1/9*11)
=1155*[(1/3*4+1/4*5+......+1/10*11)+(1/2*4+1/4*6+.......+1/8*10)+(1/1*3+1/3*5+.........+1/9*11)]
=1155*[1/3-1/4+1/4-1/5+...+1/10-1/11+1/2*(1/2-1/4)+1/2*(1/4-1/6)+....+1/2*(1/8-1/10)+1/2*(1/3-1/5)+1/2*(1/5-1/7)+..........+1/2*(1/9-1/11)]
=1155*[1/3-1/11+1/2*(1/2-1/10)+1/2*(1/3-1/11)]
=1155*[8/33+1/5+4/33]
=1155*(12/33+1/5)
=1155*12/33+1155*1/5
=420+231
=651
1/[n(n+1)]=1/n-1/(n+1)
1/[(n-1)(n+1)]=1/2*[1/(n-1)-1/(n+1)]
1155*(5/2*3*4+7/3*4*5+…+17/8*9*10+19/9*10*11)
=1155*(1/3*4+1/1*3+1/4*5+1/3*5+.........+1/9*10+1/8*10+1/10*11+1/9*11)
=1155*[(1/3*4+1/4*5+......+1/10*11)+(1/2*4+1/4*6+.......+1/8*10)+(1/1*3+1/3*5+.........+1/9*11)]
=1155*[1/3-1/4+1/4-1/5+...+1/10-1/11+1/2*(1/2-1/4)+1/2*(1/4-1/6)+....+1/2*(1/8-1/10)+1/2*(1/3-1/5)+1/2*(1/5-1/7)+..........+1/2*(1/9-1/11)]
=1155*[1/3-1/11+1/2*(1/2-1/10)+1/2*(1/3-1/11)]
=1155*[8/33+1/5+4/33]
=1155*(12/33+1/5)
=1155*12/33+1155*1/5
=420+231
=651
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