<s>[INST] Find the limit of $\frac{n}{n^3+1}+\frac{2n}{n^3+2}+\dots+\frac{n\cdot n}{n^3+n}$ 
Let $\{x_n\}_{n\geq 1}$ be defined as $x_n = \frac{n}{n^3+1}+\frac{2n}{n^3+2}+\dots+\frac{n\cdot n}{n^3+n}$. Then $\lim\limits_{n\to\infty} x_n$ is?

I want to find limit of this problem by a very specific method . I am uploading the pic of that method and my attempt.
Please guide me as to how to take this method ahead. 

 [/INST] Alternate solution:
$$x_n < \frac{n}{n^3+1} + \frac{2n}{n^3+1} + \cdots + \frac{n\cdot n}{n^3+1} = \frac{n(1+2+\cdots + n)}{n^3+1} =  \frac{n\cdot n(n+1)/2}{n^3+1}.$$
The limit of the last expression is $1/2.$ From below we have
$$\frac{n}{n^3+n} + \frac{2n}{n^3+n} + \cdots + \frac{n\cdot n}{n^3+n} = \frac{n\cdot n(n+1)/2}{n^3+n} < x_n.$$
The last fraction also has limit $1/2.$ By the squeeze theorem the desired limit is $1/2.$
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