<s>[INST] Finding the limit of sequence 
Find the limit of
  $$
\lim_{n \to \infty} \left(
    \sin \frac{n}{n^2+1^2} + 
    \sin \frac{n}{n^2+2^2} + \dotsb +
    \sin \frac{n}{n^2+n^2}
\right)
$$

I think it is a Riemann Integral but I didn't find the expression of Riemann Integral.
 [/INST] For $n\gg 0$, $\frac{n}{n^2+k^2}\approx 0$ and hence $\frac{\sin\frac{n}{n^2+k^2}}{\frac n{n^2+k^2}}\approx 1$. This allows us to forget about the sine.
Then the summands become
$$ \frac{n}{n^2+k^2}=\frac1n\cdot\frac{n^2}{n^2+k^2}=\frac1n\frac{1}{1+(k/n)^2}=\frac{f(k/n)}n$$
with $f(x)=\frac1{1+x^2}$, and there you have your Riemann sum
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