用简便方法求值:(1-1/2²)(1-1/3²)(1-1/4²)...(1-1/99²)(1-1/100²)
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(1-1/2²)(1-1/3²)(1-1/4²)...(1-1/99²)(1-1/100²)
=(1+1/2)(1-1/2)(1+1/3)(1-1/3)(1+1/4)(1-1/4).............(1+1/100)(1-1/100)
=3/2*1/2*4/3*2/3*5/4*3/4.............101/100*99/100
=1/2*101/100
=101/200
=(1+1/2)(1-1/2)(1+1/3)(1-1/3)(1+1/4)(1-1/4).............(1+1/100)(1-1/100)
=3/2*1/2*4/3*2/3*5/4*3/4.............101/100*99/100
=1/2*101/100
=101/200
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(1-1/2^2)(1-1/3^2)(1-1/4^2)…(1-1/10^2)…(1-1/n^2)
=(1-1/2)(1+1/2)(1-1/3)(1+1/3)(1-1/4)(1+1/4)(1-1/5)(1+1/5)(.....)(1-1/n)(1+1/n)
=1/2×3/2×2/3×4/3×3/4×....(1+1/n)
=1/2(1+1/n)
=1/2+1/2n
N=100
所以1/2+1/2n=1/2+1/2X100=50又1/2
采纳哦
=(1-1/2)(1+1/2)(1-1/3)(1+1/3)(1-1/4)(1+1/4)(1-1/5)(1+1/5)(.....)(1-1/n)(1+1/n)
=1/2×3/2×2/3×4/3×3/4×....(1+1/n)
=1/2(1+1/n)
=1/2+1/2n
N=100
所以1/2+1/2n=1/2+1/2X100=50又1/2
采纳哦
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=(1-1/2)(1+1/2)(1-1/3)(1+1/3)(1-1/4)(1+1/4)……(1-1/100)(1+1/100)
=1/2× 3/2×2/3 × 4/3×3/4 ×5/4×4/5 ×……×101/100
=1/2×101/100
=101/200
=1/2× 3/2×2/3 × 4/3×3/4 ×5/4×4/5 ×……×101/100
=1/2×101/100
=101/200
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原式=1*3/2^2* 2*4/3^2*..* 99*101/100^2
=101/(2*100)
=101/200
=101/(2*100)
=101/200
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