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 :) )
 A: The sum $$\sum_{i=1}^n \frac{\cos(i^2)}{n^2+i^2}$$ is in modulus less than or equal to
$$\left[ \sum_{i=1}^n \left\vert \cos(i^2) \right\vert \right] \big/ n^2$$
The numerator is less than or equal to $n$.
A: $\cos x \approx 1- \frac{x^2}{2}$ for small values of $x$. For large values of $x$ (which you need to study, since your limit is $n\to\infty$), the approximation does not hold.
To actually calculate the sum, you can notice that each element in the sum is smaller than $\frac{1}{n^2}$, and there are only $n$ elements which you sum, therefore their sum is smaller than $\frac n{n^2}=\frac1n$
A: $$\lim\limits_{n\to\infty}\left(\frac{\cos 1}{n^2 + 1}+ \frac{\cos 4}{n^2 + 4}+ \dots + \frac{\cos n^2}{n^2 + n^2}\right)$$
$$=\lim\limits_{n\to\infty}\left(\frac{\frac{\cos 1}{n^2}}{1 + \frac{1}{n^2}}+ \frac{\frac{\cos 4}{n^2}}{1 + \frac{4}{n^2}}+ \dots + \frac{\frac{\cos n^2}{n^2}}{1 + 1}\right)$$
Since the limit of each term as $n\to \infty$ yields a convergent limit, we have
$$\lim\limits_{n\to\infty} \frac{\frac{\cos 1}{n^2}}{1 + \frac{1}{n^2}}+\lim\limits_{n\to\infty} \frac{\frac{\cos 4}{n^2}}{1 + \frac{4}{n^2}}+ \dots + \lim\limits_{n\to\infty}\frac{\frac{\cos n^2}{n^2}}{1 + 1}$$
$$=\frac{0}{1 + 0}+\frac{0}{1 + 0}+ \dots + \frac{0}{1+1}=0$$
