Along the way to proving a solution for this stubborn question of mine, I've come upon this expression which I would like to evaluate: $$ \lim_{n\to\infty} \frac{1}{n^3}\sum_{\ell=1}^{n-1}\sqrt{(n^2-\ell^2)(n^2-(\ell-1)^2)} $$ Assuming consistency+correctness of the rest of my work, I would love for it to turn out that the limit is $\frac{2}{3}$, but to be honest I'm not certain of how to continue. I see no reason for there to be a nice closed form for the sum (and W|A appear to agree).
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For every $1\leqslant\ell\leqslant n-1$, $$ n^2-\ell^2\leqslant\sqrt{(n^2-\ell^2)(n^2-(\ell-1)^2)}\leqslant n^2-(\ell-1)^2, $$ hence the sums $S_n$ you are interested in are such that $R_n\leqslant S_n\leqslant T_n$ for every $n\geqslant1$, with $$ R_n=\frac1{n^3}\sum_{\ell=1}^{n-1}(n^2-\ell^2),\qquad T_n=\frac1{n^3}\sum_{\ell=0}^{n-2}(n^2-\ell^2). $$ The rest should be easy (and the limit is indeed $\frac23$).
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1$\begingroup$ Excellent! And, thrilled to see that the solution is correct. I'm going to write up an answer to the old question now. $\endgroup$ Aug 13, 2012 at 8:59
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2$\begingroup$ I wonder the question @did can not solve... $\endgroup$ Aug 13, 2012 at 9:19
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$\begingroup$ Here is what I ended up doing with this. Thanks again! $\endgroup$ Aug 13, 2012 at 9:41