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(1^4/4)(3^4+1/4)...(19^4+1\4) \ (2^4+1/4)(2^4+1/4)(4^4+1/4).

来源:学生作业帮 编辑:作业帮 分类:数学作业 时间:2024/03/29 08:57:48
(1^4/4)(3^4+1/4)...(19^4+1\4) \ (2^4+1/4)(2^4+1/4)(4^4+1/4)...(20^4+1/4)=?
分子第一项:(1^4+1/4)
x^4+1/4=x^4+x^2+1/4-x^2=(x^2+1/2)^2-x^2)
=(x^2+1/2+x)(x^2+1/2-x)=(x+1/2)^2+1/4)(x-1/2)^2+1/4)
即得x^4+1/4=(x+1/2)^2+1/4)(x-1/2)^2+1/4)
(1^4+1/4)*(3^4+1/4)*.(19^4+1/4)/(2^4+1/4)*(4^4+1/4).(20^4+1/4)
={((1/2)^2+1/4)((3/2)^2+1/4)((5/2)^2+1/4)((7/2)^2+1/4)...((37/2)^2+1/4)((39/2)^2+1/4)}/{((3/2)^2+1/4)((5/2)^2+1/4)((7/2)^2+1/4)((9/2)^2+1/4)...((39/2)^2+1/4)((41/2)^2+1/4)}
=((1/2)^2+1/4)/((41/2)^2+1/4)=2/(41^2+1)=1/841