<s>[INST] Can someone explain the limit $\lim _{n \rightarrow \infty} \left(\frac{1}{\sqrt{n^2+1}}+\frac{1}{\sqrt{n^2+2}}+\cdots+\frac{1}{\sqrt{n^2+n}}\right)$? $$\begin{aligned}
&\text { Find the following limit: } \lim _{n \rightarrow \infty} \left(\frac{1}{\sqrt{n^2+1}}+\frac{1}{\sqrt{n^2+2}}+\cdots+\frac{1}{\sqrt{n^2+n}}\right)\\
&\text { **In a solution I got** Ans:- Let } u_n=\frac{n}{\sqrt{n^2+n}}\\
&\therefore \lim _{n \rightarrow \infty} u_n=\lim _{n \rightarrow \infty} \frac{n}{\sqrt{n^{2}+n}}=\lim _{n \rightarrow \infty} \frac{1}{\sqrt{1+\frac{1}{n}}}=1\\
&\text { By Cauchy's first theorem:- } \lim _{n \rightarrow \infty} \left(\frac{u_1+\cdots+u_n}{n}\right)=1\\
&\text{So, } \lim _{n \rightarrow \infty}\left(\frac{1}{\sqrt{n^2+1}}+\frac{1}{\sqrt{n^2+2}}+\cdots+\frac{1}{\sqrt{n^2+n}}\right)=1
\end{aligned}$$
I am unable to understand this. How the general term comes like this! Any such explanation to the whole problem would be greatly appreciated.
 [/INST] You cannot apply Cauchy's first theorem   because
$$\sum_{k=1}^n\frac{1}{\sqrt{n^2+k}}\ne \frac{\sum_{k=1}^nu_k}{n}$$
since
$$\sum_{k=1}^nu_k=\sum_{k=1}^n\frac{k}{\sqrt{k^2+k}}$$
</s>