Which is the limit of the sequence $\sum_{i=1}^n \frac{\cos(i^2)}{n^2+i^2}$ I can't find the limit as n $\to$ infinity of the sequence:
$$\frac{\cos(1)}{n^2 + 1} + \frac{\cos(4)}{n^2 + 4} + \dots + \frac{\cos(n^2)}{n^2 + n^2}$$
I tried to use the inequality $\cos(n^2) < n$ which I could justify it by using Taylor series of $\cos x$ and replacing the $x$ with $x^2$ but my professor told me I was wrong.Plus I plotted $\cos(x^2)$ and $x$ and I saw that he was right. So I don't have any ideas left and I need some help here. Even though this is an exercise that my professor gave me it is not an exercise for the semester or anything like that it's more like exercise that he gave me for fun. (I asked for it.I should have been more careful :) )

$$\lim\limits_{n\to\infty}\left(\frac{\cos 1}{n^2 + 1}+ \frac{\cos 4}{n^2 + 4}+ \dots + \frac{\cos n^2}{n^2 + n^2}\right)$$
$$=\lim\limits_{n\to\infty}\left(\frac{\frac{\cos 1}{n^2}}{1 + \frac{1}{n^2}}+ \frac{\frac{\cos 4}{n^2}}{1 + \frac{4}{n^2}}+ \dots + \frac{\frac{\cos n^2}{n^2}}{1 + 1}\right)$$
Since the limit of each term as $n\to \infty$ yields a convergent limit, we have
$$\lim\limits_{n\to\infty} \frac{\frac{\cos 1}{n^2}}{1 + \frac{1}{n^2}}+\lim\limits_{n\to\infty} \frac{\frac{\cos 4}{n^2}}{1 + \frac{4}{n^2}}+ \dots + \lim\limits_{n\to\infty}\frac{\frac{\cos n^2}{n^2}}{1 + 1}$$
$$=\frac{0}{1 + 0}+\frac{0}{1 + 0}+ \dots + \frac{0}{1+1}=0$$
