study of two sequences I need to study wether thos two sequences converge or not.
1) $u_n=n\sum_{k=1}^{2n+1} \frac{1}{n^2+k}$
2) $v_n=\frac{1}{n}\sum_{k=0}^{n-1} \cos(\frac{1}{\sqrt{n+k}})$
For the first i get it converges to $0$ by just expressing the sum as a fraction (not very smart imo).

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*Note that $(n^{2} +k) \leq (n^{2}+2n+1)$ whenever $k \in\left\{1,2,\cdots,(2n+1)\right\}$. So we have that $\frac{1}{n^{2}+k} \geq \frac{1}{n^{2}+2n+1} =\frac{1}{(n+1)^{2}}$. Similarly $n^{2}+1 \leq n^{2}+k$ and so we have that $\frac{1}{n^{2}+1}\geq \frac{1}{n^2+k}$. So we get that $$\sum_{k=1}^{2n+1} \frac{1}{n^{2}+k} \geq \sum_{k=1}^{2n+1} \frac{1}{n^{2}+2n+1} \ \ \mbox{and} \ \ \sum_{k=1}^{2n+1} \frac{1}{n^{2}+k} \leq \sum_{k=1}^{2n+1}\frac{1}{n^{2}+1}$$ Using this we have $$ \frac{2n^{2}}{n^{2}+2n+1} \leq u_{n} \leq \frac{2n^{2}}{n^{2}+1}$$ Now apply the Squeeze Theorem.

