1/1乘2乘3+1/2乘3乘4+...+1/98乘99乘100=
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1/(1*2*3)+1/(2*3*4)+...+1/(98*99*100)
=(1/2)[1/(1*2)-1/(2*3)]+(1/2)[1/(2*3)-1/(3*4)]+...+(1/2)[1/(98*99)-1/(99*100)]
=(1/2)[1/(1*2)]-(1/2)[1/(2*3)]+(1/2)[1/(2*3)]-(1/2)[1/(3*4)]+...+(1/2)[1/(98*99)]-(1/2)[1/(99*100)]
中间的全部互相抵消，所以
=(1/2)[1/(1*2)]-(1/2)[1/(99*100)]
=4949/19800
=(1/2)[1/(1*2)-1/(2*3)]+(1/2)[1/(2*3)-1/(3*4)]+...+(1/2)[1/(98*99)-1/(99*100)]
=(1/2)[1/(1*2)]-(1/2)[1/(2*3)]+(1/2)[1/(2*3)]-(1/2)[1/(3*4)]+...+(1/2)[1/(98*99)]-(1/2)[1/(99*100)]
中间的全部互相抵消，所以
=(1/2)[1/(1*2)]-(1/2)[1/(99*100)]
=4949/19800
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