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求和:(1/2²-1)+(1/3²-1)+(1/4²-1)+.+(1/n²-1)
题目详情
求和:(1/2²-1)+(1/3²-1)+(1/4²-1)+.+(1/n²-1)
▼优质解答
答案和解析
∵1/﹙n²-1﹚=2/﹙n-1﹚-2/﹙n+1﹚
∴(1/2²-1)+(1/3²-1)+(1/4²-1)+.+(1/n²-1)
=(2/1×3)+(2/2×4)+(2/3×5)+.+﹛2/﹙n-1)﹙n+1﹚﹜
=2/1-2/3+2/2-2/4+2/3-2/5+.+2/﹙n-1)-2/﹙n+1﹚
=2+1－2/n－2/﹙n+1﹚
=﹙3n²+n-1﹚／[n﹙n+1﹚]
∴(1/2²-1)+(1/3²-1)+(1/4²-1)+.+(1/n²-1)
=(2/1×3)+(2/2×4)+(2/3×5)+.+﹛2/﹙n-1)﹙n+1﹚﹜
=2/1-2/3+2/2-2/4+2/3-2/5+.+2/﹙n-1)-2/﹙n+1﹚
=2+1－2/n－2/﹙n+1﹚
=﹙3n²+n-1﹚／[n﹙n+1﹚]
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