<s>[INST] Evaluating limit by sandwich theorem: $\lim_{n\to\ \infty}\frac{1}{1+n^2} +\frac{2}{2+n^2}+\cdots+\frac{n}{n+n^2}$ $$\lim_{n\to\ \infty}\frac{1}{1+n^2} +\frac{2}{2+n^2}+\cdots+\frac{n}{n+n^2}$$
For using Sandwich theorem I need two functions such that $g(x)<f(x)<h(x)$
$$\frac{1}{n+n^2} +\frac{2}{n+n^2}+\cdots+\frac{n}{n+n^2} \leq \frac{1}{1+n^2} +\frac{2}{2+n^2}+\cdots+\frac{n}{n+n^2} $$
But I can't find function greater than the given which will help me evaluate the limit.
 [/INST] HINT:
If you are familiar with the harmonic numbers, you can also proof:
$$\lim_{n\to\infty}\frac{1}{1+n^2}+\frac{2}{2+n^2}+\cdots+\frac{n}{n+n^2}=\lim_{n\to\infty}\sum_{m=1}^{n}\frac{m}{m+n^2}=$$
$$\lim_{n\to\infty}n\left(n\text{H}_{n^2}-n\text{H}_{n^2+n}+1\right)=\frac{1}{2}$$
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