lim(1/(a1×a2)+1/(a2×a3)+……+1/(an+a(n+1))=( ) 该数列Sn=2n²+
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lim(1/(a1×a2)+1/(a2×a3)+……+1/(an+a(n+1))=( ) 该数列Sn=2n²+n
![lim(1/(a1×a2)+1/(a2×a3)+……+1/(an+a(n+1))=( ) 该数列Sn=2n²+](/uploads/image/z/16088307-51-7.jpg?t=lim%281%2F%28a1%C3%97a2%29%2B1%2F%28a2%C3%97a3%29%2B%E2%80%A6%E2%80%A6%2B1%2F%EF%BC%88an%2Ba%28n%2B1%29%29%3D%28+%29+%E8%AF%A5%E6%95%B0%E5%88%97Sn%3D2n%26%23178%3B%2B)
n=1时,a1=S1=2×1²+1=3
n≥2时,Sn=2n²+n S(n-1)=2(n-1)²+(n-1)
an=Sn-S(n-1)=2n²+n-2(n-1)²-(n-1)=4n-1
n=1时,a1=4-1=3,同样满足.
数列{an}的通项公式为an=4n-1
1/[ak·a(k+1)]=1/[(4n-1)(4(n+1)-1]=(1/4)[1/(4n-1) -1/(4(n+1)-1]
1/(a1·a2)+1/(a2·a3)+...+1/[an·a(n+1)]
=(1/4)[1/(4×1-1)-1/(4×2-1)+1/(4×2-1)-1/(4×3-1)+...+1/(4n-1)-1/(4(n+1)-1)]
=(1/4)[1/3 -1/(4n+3)
=4/3 -1/[4(4n+3)]
n->+∞,4n+3->+∞ 4(4n+3)->+∞,1/[4(4n+3)]->0
4/3 -1/[4(4n+3)]->4/3
lim[1/(a1·a2)+1/(a2·a3)+...+1/[an·a(n+1)]] =4/3
n≥2时,Sn=2n²+n S(n-1)=2(n-1)²+(n-1)
an=Sn-S(n-1)=2n²+n-2(n-1)²-(n-1)=4n-1
n=1时,a1=4-1=3,同样满足.
数列{an}的通项公式为an=4n-1
1/[ak·a(k+1)]=1/[(4n-1)(4(n+1)-1]=(1/4)[1/(4n-1) -1/(4(n+1)-1]
1/(a1·a2)+1/(a2·a3)+...+1/[an·a(n+1)]
=(1/4)[1/(4×1-1)-1/(4×2-1)+1/(4×2-1)-1/(4×3-1)+...+1/(4n-1)-1/(4(n+1)-1)]
=(1/4)[1/3 -1/(4n+3)
=4/3 -1/[4(4n+3)]
n->+∞,4n+3->+∞ 4(4n+3)->+∞,1/[4(4n+3)]->0
4/3 -1/[4(4n+3)]->4/3
lim[1/(a1·a2)+1/(a2·a3)+...+1/[an·a(n+1)]] =4/3
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