计算1/1×3+1/3×5+1/5×7...+1/2009×2011

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计算1/1×3+1/3×5+1/5×7...+1/2009×2011计算1/1×3+1/3×5+1/5×7...+1/2009×2011计算1/1×3+1/3×5+1/5×7...+1/2009×20

计算1/1×3+1/3×5+1/5×7...+1/2009×2011
计算1/1×3+1/3×5+1/5×7...+1/2009×2011

计算1/1×3+1/3×5+1/5×7...+1/2009×2011
1/1×3=1/2×(1/1-1/3)
1/3×5=1/2×(1/3-1/5)
1/5×7=1/2×(1/5-1/7)
.
1/2009×2011=1/2×(1/2009-1/2011)
所以
原式=1/2×(1-1/3+1/3-1/5+...+1/2009-1/2011)
=1/2×(1-1/2011)
=1005/2011

1/(1*3)=(1/2)(1-1/3)
1/3×5=(1/2)(1/3-1/5)
...
1/2009×2011=(1/2)(1/2009-1/2011)
所以原式=(1/2)(1-1/3+1/3-1/5+...+1/2009-1/2011)
=(1/2)(1-1/2011)=1005/2011

原式=(1-1/3+1/3-1/5+1/5-1/7……+1/2009-1/2011)/2
=(1-1/2011)/2
=(2010/2011)/2
=1005/2011