<s>[INST] Find $\lim_{n \to \infty} \left(n - \sum_{k=1} ^{n} \cos \frac{\sqrt{k}}{n} \right)$ Find $\lim_{n \to \infty} \left(n - \sum_{k=1} ^{n} \cos \frac{\sqrt{k}}{n} \right)$
My Attempt:
$\forall x: \ |\cos(x)|\leq 1$, Therefore:
$$\lim_{n \to \infty} \left(n - \sum_{k=1} ^{n} \cos \frac{\sqrt{k}}{n} \right) \leq \lim_{n \to \infty} \left(n - \left| \sum_{k=1} ^{n} \cos \frac{\sqrt{k}}{n}  \right| \right) \leq  \lim_{n \to \infty} \left( \sum_{k=1}^{n} |\cos \frac{\sqrt{k}}{n} | \right) \leq \lim_{n \to \infty} \left(n - \sum_{k=1} ^{n} k \right) = 0$$
but I can't find a way to bound the limit such that I could prove that:
$$0 \leq \lim_{n \to \infty} \left(n - \sum_{k=1} ^{n} \cos \frac{\sqrt{k}}{n} \right) \leq 0$$ 
which would finish the proof. 
 [/INST] You have
$$u_n = n - \sum_{k=1} ^{n} \cos \frac{\sqrt{k}}{n} = \sum_{k=1} ^{n}\left(1-\cos \frac{\sqrt{k}}{n}\right)$$
And as $\sum_{k=1} ^{n} k = \frac{n(n+1)}{2}$:
$$u_n - \frac{n+1}{4n} = \sum_{k=1} ^{n}\left(1-\cos \frac{\sqrt{k}}{n}-\frac{k}{2n^2}\right )$$
And using Taylor's theorem, you get for all $x \in \mathbb R$
$$\left\vert 1-\cos(x)-x^2/2 \right\vert \leq x^4/24.$$
Therefore
$$\left\vert u_n - \frac{n+1}{4n} \right\vert \le \sum_{k=1} ^{n}\left\vert1-\cos \frac{\sqrt{k}}{n}-\frac{k}{4n^2}\right\vert \le \frac{1}{24n^4}\sum_{k=1} ^{n} k^2 \tag{1}$$
As $$\sum_{k=1} ^{n} k^2 = \frac{n(n+1)(2n+1)}{6}$$ the RHS of inequality $(1)$ converges to $0$.
Finaly $(u_n)$ converges to $1/4$ as $\lim\limits_{n \to \infty} \frac{n+1}{4n} =1/4$.
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