1/2+1/6+1/12+1/20+1/30+1/42+1/56=1-1/2+1/2-1/3+1/3-1/4+1/4-1/5+1/5-1/6+1/6-1/7+1/7-1/8=1-1/8=7/8这一步=1-1/2+1/2-1/3+1/3-1/4+1/4-1/5+1/5-1/6+1/6-1/7+1/7-1/8怎么会得到这步=1-1/8
来源:学生作业帮助网 编辑:作业帮 时间:2024/08/30 05:39:25
![1/2+1/6+1/12+1/20+1/30+1/42+1/56=1-1/2+1/2-1/3+1/3-1/4+1/4-1/5+1/5-1/6+1/6-1/7+1/7-1/8=1-1/8=7/8这一步=1-1/2+1/2-1/3+1/3-1/4+1/4-1/5+1/5-1/6+1/6-1/7+1/7-1/8怎么会得到这步=1-1/8](/uploads/image/z/9282681-9-1.jpg?t=1%2F2%2B1%2F6%2B1%2F12%2B1%2F20%2B1%2F30%2B1%2F42%2B1%2F56%3D1-1%2F2%2B1%2F2-1%2F3%2B1%2F3-1%2F4%2B1%2F4-1%2F5%2B1%2F5-1%2F6%2B1%2F6-1%2F7%2B1%2F7-1%2F8%3D1-1%2F8%3D7%2F8%E8%BF%99%E4%B8%80%E6%AD%A5%3D1-1%2F2%2B1%2F2-1%2F3%2B1%2F3-1%2F4%2B1%2F4-1%2F5%2B1%2F5-1%2F6%2B1%2F6-1%2F7%2B1%2F7-1%2F8%E6%80%8E%E4%B9%88%E4%BC%9A%E5%BE%97%E5%88%B0%E8%BF%99%E6%AD%A5%3D1-1%2F8)
1/2+1/6+1/12+1/20+1/30+1/42+1/56=1-1/2+1/2-1/3+1/3-1/4+1/4-1/5+1/5-1/6+1/6-1/7+1/7-1/8=1-1/8=7/8这一步=1-1/2+1/2-1/3+1/3-1/4+1/4-1/5+1/5-1/6+1/6-1/7+1/7-1/8怎么会得到这步=1-1/8
1/2+1/6+1/12+1/20+1/30+1/42+1/56
=1-1/2+1/2-1/3+1/3-1/4+1/4-1/5+1/5-1/6+1/6-1/7+1/7-1/8
=1-1/8
=7/8
这一步=1-1/2+1/2-1/3+1/3-1/4+1/4-1/5+1/5-1/6+1/6-1/7+1/7-1/8
怎么会得到这步=1-1/8
1/2+1/6+1/12+1/20+1/30+1/42+1/56=1-1/2+1/2-1/3+1/3-1/4+1/4-1/5+1/5-1/6+1/6-1/7+1/7-1/8=1-1/8=7/8这一步=1-1/2+1/2-1/3+1/3-1/4+1/4-1/5+1/5-1/6+1/6-1/7+1/7-1/8怎么会得到这步=1-1/8
56=7*8=7*2*2*2
42=7*6=7*2*3
30=5*6=5*2*3
20=5*4=5*2*2
12=3*4=3*2*2
6=2*3
1/56=1/7*1/2*1/2*1/2
1/42=1/7*1/2*1/3
1/30=1/5*1/2*1/3
1/20=1/5*1/2*1/2
1/12=1/3*1/2*1/2
1/6=1/3*1/2
1/2+1/6+1/12+1/20+1/30+1/42+1/56
=1/2+1/3*1/2+1/3*1/2*1/2+1/5*1/2*1/2+1/5*1/2*1/3+1/7*1/2*1/3+1/7*1/2*1/2*1/2=就这样往下推,得出答案,竟然一分都没给,被你骗了,我新手,没注意过分数,做完了才看到你的是零蛋!
1/2+1/6+1/12+1/20+1/30+1/42+1/56
=1/(1×2)+1/(2×3)+1/(3×4)+1/(4×5)+1/(5×6)+1/(6×7) +1/(7×8)
=(1-1/2)+(1/2-1/3)+(1/3-1/4)+(1/4-1/5)+(1/5-1/6)+(1/6-1/7)+(1/7-1/8)
=1-1/2+1/2-1/3+1/3-1/4+1/4-...
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1/2+1/6+1/12+1/20+1/30+1/42+1/56
=1/(1×2)+1/(2×3)+1/(3×4)+1/(4×5)+1/(5×6)+1/(6×7) +1/(7×8)
=(1-1/2)+(1/2-1/3)+(1/3-1/4)+(1/4-1/5)+(1/5-1/6)+(1/6-1/7)+(1/7-1/8)
=1-1/2+1/2-1/3+1/3-1/4+1/4-1/5+1/5-1/6+1/6-1/7+1/7-1/8
=1+(-1/2+1/2-1/3+1/3-1/4+1/4-1/5+1/5-1/6+1/6-1/7+1/7)-1/8
=1-1/8
收起
这叫"裂项"
可以证明这一规律:1/n(n+1)=1/n-1/(n+1
那一步中,中间相邻的两项都互为相反数,一正一负就都消掉了.
因为1/2=1-1/2,1/6=1/2-1/3,1/12=1/3-1/4,1/20=1/4-1/5,1/30=1/5-1/6,1/42=1/6-1/7,1/56=1/7-1/8,换算后为1-1/8。