<s>[INST] how can I show that the sequence $x_{n}=\frac{n}{n^{2}+1}+\frac{n}{n^2+2}+...+\frac{n}{n^2+n},n=1,2,...$ converges? How can I check if the sequence $x_{n}=\frac{n}{n^{2}+1}+\frac{n}{n^2+2}+...+\frac{n}{n^2+n},n=1,2,...$ converges?Is there a theorem that I could use?
 [/INST] Notice the following:
Every two integers $n,k$ such that $1\leq k \leq n$  satisfy 
$$\frac {1}{n+1} = \frac{n}{n^2+n} \leq \frac{n}{n^2+k} \leq \frac{n}{n^2}=\frac{1}{n}$$
So we get 
$$ \frac{1}{1+\frac{1}{n}} = n \cdot \frac {1}{n+1} \leq x_n \leq n \cdot \frac{1}{n} =1$$
Thus by the Sandwich Theorem we get $x_n\rightarrow 1$.
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