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(1- 1/2²)(1- 1/3²)...(1- 1/2014²)(1- 1/2015²)
=(1- 1/2)(1+ 1/2)(1- 1/3)(1+ 1/3)...(1- 1/2014)(1+ 1/2014)(1- 1/2015)(1+ 1/2015)
=(1/2)(3/2)(2/3)(4/3)...(2013/2014)(2015/2014)(2014/2015)(2016/2015)
=[(1/2)(2/3)...(2013/2014)(2014/2015)][(3/2)(4/3)...(2015/2014)(2015/2015)]
=[(1×2×...×2013×2014)/(2×3×...×2014×2015)][(3×4×...×2015×2016)/(2×3×...×2014×2015)]
=(1/2015)×(2016/2)
=1008/2015
=(1- 1/2)(1+ 1/2)(1- 1/3)(1+ 1/3)...(1- 1/2014)(1+ 1/2014)(1- 1/2015)(1+ 1/2015)
=(1/2)(3/2)(2/3)(4/3)...(2013/2014)(2015/2014)(2014/2015)(2016/2015)
=[(1/2)(2/3)...(2013/2014)(2014/2015)][(3/2)(4/3)...(2015/2014)(2015/2015)]
=[(1×2×...×2013×2014)/(2×3×...×2014×2015)][(3×4×...×2015×2016)/(2×3×...×2014×2015)]
=(1/2015)×(2016/2)
=1008/2015
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