相邻分母间的差值依次为4,6,8,10,...,198.
由此可知上式共有99项
所以
1/2+5/6+11/12+19/20+29/30+......+9701/9702+9899/9900=
99-(1/2+1/6+1/12+1/20+1/30+......+1/9702+1/9900)=
99-(1/2+(1/2-1/3)+(1/3-1/4)+(1/4-1/5)+(1/5-1/6)+......+(1/98-1/99)+(1/99-1/100))=
99-(1/2+1/2-1/100)=
9801/100
由此可知上式共有99项
所以
1/2+5/6+11/12+19/20+29/30+......+9701/9702+9899/9900=
99-(1/2+1/6+1/12+1/20+1/30+......+1/9702+1/9900)=
99-(1/2+(1/2-1/3)+(1/3-1/4)+(1/4-1/5)+(1/5-1/6)+......+(1/98-1/99)+(1/99-1/100))=
99-(1/2+1/2-1/100)=
9801/100
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