lim(1/n²+1/(n+1)²+………1/(2n²)）=0求解
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n>1时，
1/n²+1/(n+1)²+………+1/(2n)²
<1/[(n-1)n]+1/[n(n+1)]+.....+1/[(2n-1)2n]
=1/(n-1)-1/n+1/n-1/(n+1)+....+1/(2n-1)-1/(2n)
=1/(n-1)-1/(2n)
=(n+1)/[2n(n-1)]
=(1+1/n)*1/(2n-2)
1/n²+1/(n+1)²+………+1/(2n)²
> 1/[n(n+1)]+1/[(n+1)(n+2)]+....+1/[2n(2n+1)]
=1/n-1/(n+1)+1/(n+1)-1/(n+2)+....+1/(2n)-1/(2n+1)
=1/n-1/(2n+1)
=(n+1)/[n(2n+1)]
=(1+1/n)*1/(2n+1)
∵(1+1/n)*1/(2n-2)与=(1+1/n)*1/(2n+1)的极限均为0
∴lim(1/n²+1/(n+1)²+………1/(2n)²）=0
1/n²+1/(n+1)²+………+1/(2n)²
<1/[(n-1)n]+1/[n(n+1)]+.....+1/[(2n-1)2n]
=1/(n-1)-1/n+1/n-1/(n+1)+....+1/(2n-1)-1/(2n)
=1/(n-1)-1/(2n)
=(n+1)/[2n(n-1)]
=(1+1/n)*1/(2n-2)
1/n²+1/(n+1)²+………+1/(2n)²
> 1/[n(n+1)]+1/[(n+1)(n+2)]+....+1/[2n(2n+1)]
=1/n-1/(n+1)+1/(n+1)-1/(n+2)+....+1/(2n)-1/(2n+1)
=1/n-1/(2n+1)
=(n+1)/[n(2n+1)]
=(1+1/n)*1/(2n+1)
∵(1+1/n)*1/(2n-2)与=(1+1/n)*1/(2n+1)的极限均为0
∴lim(1/n²+1/(n+1)²+………1/(2n)²）=0
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