2²/1×3+4²/3×5+6²/5×7+8²/7×9+……+2010²/2009×2011

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2²/1×3+4²/3×5+6²/5×7+8²/7×9+……+2010²/2009×20112²/1×3+4²/3×5+6

2²/1×3+4²/3×5+6²/5×7+8²/7×9+……+2010²/2009×2011
2²/1×3+4²/3×5+6²/5×7+8²/7×9+……+2010²/2009×2011

2²/1×3+4²/3×5+6²/5×7+8²/7×9+……+2010²/2009×2011
2²/1×3+4²/3×5+6²/5×7+8²/7×9+……+2010²/2009×2011
=2^2/(2^2-1)+4^2/(4^2-1)+6^2/(6^2-1)+...+2010^2/(2010^2-1)
=1+1/(2^2-1)+1+1/(4^2-1)+...+1+1/(2010^2-1)
=1005+1/1*3+1/3*5+1/5*7+...+1/2009*2011
=1005+1/2(1/1-1/3+1/3-1/5+...+1/2009-1/2011)
=1005+1/2(1-1/2011)
=1005又1005/2011