√((1+1/N^2+1/(N+1)^2)

来源：百度知道编辑：UC知道时间：2024/07/09 02:19:57

1.化简√((1+1/N^2+1/(N+1)^2)的结果
2.根据1.的结果计算√(1+1/1^2+1/2^2)+√(1+1/2^2+1/3^2)+√(1+1/1^3+1/4^2)+…√(1+1/2007^2+1/2008^2)

1+1/N^2+1/(N+1)^2
=[N^2(N+1)^2+(N+1)^2+N^2]/N^2(N+1)^2

分子=N^2(N+1)^2+2N(N+1)+[(N+1)^2-2N(N+1)+N^2]
=N^2(N+1)^2+2N(N+1)+(N+1-N)^2
=N^2(N+1)^2+2N(N+1)+1
=[N(N+1)+1]^2
=(N^2+N+1)^2

分母=(N^2+N)^2

所以√((1+1/N^2+1/(N+1)^2)=(N^2+N+1)/(N^2+N)
=1+1/N(N+1)=1+1/N-1/(N+1)

√(1+1/1^2+1/2^2)+√(1+1/2^2+1/3^2)+√(1+1/1^3+1/4^2)+…√(1+1/2007^2+1/2008^2)
=(1+1/1-1/2)+(1+1/2-1/3)+……+(1+1/2007-1/2008)
=1+1+……+1(2007个)+(1-1/2+1/2-1/3+……+1/2007-1/2008)
=2007+1-1/2008
=2007+2007/2008
=2007又2007/2008

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