1/1×2×3×4+1/2×3×4×5+1/3×4×5×6……1/7×8×9×10=？
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因为：1/n(n+1)(n+2)(n+3)
=1/[n(n+3)][(n+1)(n+2)]
=1/[(n²+3n)(n²+3n+3)]
=(1/3)[1/n(n+3)-1/(n+1)(n+2)]
=(1/3){(1/3)·[(1/n)-1/(n+3)]-[1/(n+1)-1/(n+2)]}
=(1/9)·[(1/n)-(1/n+3)]-(1/3)[1/(n+1)-1/(n+2)]
所以原式=(1/9)·[1-(1/4)+(1/2)-(1/5)+(1/3)-(1/6)+(1/4)-(1/7)+……+(1/7)-(1/10)]-(1/3)·[(1/2)-(1/3)+(1/3)-(1/4)+……+(1/7)-(1/8)+(1/8)-(1/9)]
=(1/9)·[1+(1/2)+(1/3)-(1/8)-(1/9)-(1/10)]-(1/3)·[(1/2)-(1/9)]
=……
=119/3240
=1/[n(n+3)][(n+1)(n+2)]
=1/[(n²+3n)(n²+3n+3)]
=(1/3)[1/n(n+3)-1/(n+1)(n+2)]
=(1/3){(1/3)·[(1/n)-1/(n+3)]-[1/(n+1)-1/(n+2)]}
=(1/9)·[(1/n)-(1/n+3)]-(1/3)[1/(n+1)-1/(n+2)]
所以原式=(1/9)·[1-(1/4)+(1/2)-(1/5)+(1/3)-(1/6)+(1/4)-(1/7)+……+(1/7)-(1/10)]-(1/3)·[(1/2)-(1/3)+(1/3)-(1/4)+……+(1/7)-(1/8)+(1/8)-(1/9)]
=(1/9)·[1+(1/2)+(1/3)-(1/8)-(1/9)-(1/10)]-(1/3)·[(1/2)-(1/9)]
=……
=119/3240
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