1/1×3+1/3×5+1/5×7+···+1/(2n-1)(2n+1)

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1/1×3+1/3×5+1/5×7+···+1/(2n-1)(2n+1)

1/1×3+1/3×5+1/5×7+···+1/(2n-1)(2n+1)
1/1×3+1/3×5+1/5×7+···+1/(2n-1)(2n+1)

1/1×3+1/3×5+1/5×7+···+1/(2n-1)(2n+1)
1/﹙1×3﹚+1/﹙3×5﹚+1/﹙5×7﹚+……+1/﹙2n-1﹚﹙2n+1﹚
=1/2×[1-1/3+1/3-1/5+1/5-1/7+……+1/﹙2n-1﹚-1/﹙2n+1﹚]
=1/2×[1-1/﹙2n+1﹚]
=1/2×2n/﹙2n+1﹚
=n/﹙2n+1﹚.

1/1×3+1/3×5+1/5×7+···+1/(2n-1)(2n+1)
=1/2x[1-1/3+1/3-1/5+...+1/(2n-1)-1/(2n+1)]
=1/2x[1-1/(2n+1)]
=n/(2n+1)