<s>[INST] How can I solve the following sequence Let we have the following sequence 
$$y_n= \frac{1}{\sqrt{n^2+1}}+\frac{1}{\sqrt{n^2+2}}+\cdots+\frac{1}{\  \sqrt{n^2+n}}$$ find the limit of the sequence $y_n$ decide whether it is increasing or decreasing 
 [/INST] Hint 
Prove that $$\frac{n}{\sqrt{n^2+n}}\leq \sum\frac{1}{\sqrt{n^2+k}}\leq\frac{n}{\sqrt{n^2}}$$
$$\frac{n}{\sqrt{n^2+n}}\leq y_n\leq\frac{n}{\sqrt{n^2}}$$
Then use squeeze lemma
It is a increasing sequence
Knowing that $$\frac{n}{\sqrt{n^2+n}}\leq y_n\leq\frac{1}{n}$$
We have that $y_n\leq\frac{1}{n}$ and $\frac{n+1}{\sqrt{(n+1)^2+n+1}}\leq y_{n+1}$
And easy you can show that $$\frac{1}{n}\leq \frac{n+1}{\sqrt{(n+1)^2+n+1}}$$ 
So $$y_
n\leq y_{n+1}$$
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