<s>[INST] limit of sum defined sequence Let $x_n=\displaystyle \sum_{k=1}^n \sqrt{1+\frac{k}{n^2}}, n\ge1$. Prove that $\displaystyle \lim_{n \rightarrow \infty} n (x_n-n-\frac{1}{4})=\frac{5}{24}$.
What I've done:it's easy to show that $\displaystyle \lim_{n \rightarrow \infty} \frac{x_n}{n}=1$ and $\displaystyle \lim_{n \rightarrow \infty} (x_n-n)=\frac{1}{4}$
 [/INST] Expand the square root in a taylor series to three terms, to get $1+k/2n^2 -k^2/8 n^4+ k^3/16 n^6,$ then sum.
The sum of the $1$s gives you $n$. The sum of $k/n^2$ gives $n(n+1)/4n^2,$ the sum of $k^2/n^4$ gives you $(1+n)(2+n)/48 n^3,$ and the next term is $O(1/n^2).$
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