1^2+2^2+...+n^2=(1/6)n(n+1)(2n+1)
1^2/(n^3+n^2+n+1) +2^2/(n^3+n^2+n+1)+...+n^2/(n^3+n^2+n+n)
>(1/6)n(n+1)(2n+1)/(n^3+n^2+n+n)
and
1^2/(n^3+n^2+n+1) +2^2/(n^3+n^2+n+1)+...+n^2/(n^3+n^2+n+n)
<(1/6)n(n+1)(2n+1)/(n^3+n^2+n+1)
lim(n->∞) (1/6)n(n+1)(2n+1)/(n^3+n^2+n+n) = 1/3
lim(n->∞) (1/6)n(n+1)(2n+1)/(n^3+n^2+n+1) = 1/3
=>
lim(n->∞) [1^2/(n^3+n^2+n+1) +2^2/(n^3+n^2+n+1)+...+n^2/(n^3+n^2+n+n)] =1/3
1^2/(n^3+n^2+n+1) +2^2/(n^3+n^2+n+1)+...+n^2/(n^3+n^2+n+n)
>(1/6)n(n+1)(2n+1)/(n^3+n^2+n+n)
and
1^2/(n^3+n^2+n+1) +2^2/(n^3+n^2+n+1)+...+n^2/(n^3+n^2+n+n)
<(1/6)n(n+1)(2n+1)/(n^3+n^2+n+1)
lim(n->∞) (1/6)n(n+1)(2n+1)/(n^3+n^2+n+n) = 1/3
lim(n->∞) (1/6)n(n+1)(2n+1)/(n^3+n^2+n+1) = 1/3
=>
lim(n->∞) [1^2/(n^3+n^2+n+1) +2^2/(n^3+n^2+n+1)+...+n^2/(n^3+n^2+n+n)] =1/3