<s>[INST] How can I check the convergence of the sequence? Does it diverge? How can I check the convergence of the sequence $\frac{1}{\sqrt{n^2+1}}+\frac{2}{\sqrt{n^2+2}}+\cdots+\frac{n}{\sqrt{n^2+n}}$? I think that it diverges,because it is bounded below from $\frac{n(n+1)}{2\sqrt{n^2+n}} $ and above from $\frac{n(n+1)}{2\sqrt{n^2+1}}$..Is this correct?
 [/INST] HINT: you can use Squeeze Theorem.
$\frac{1}{\sqrt{n^2+n}}+\frac{2}{\sqrt{n^2+n}}+\cdots+\frac{n}{\sqrt{n^2+n}}\leq\frac{1}{\sqrt{n^2+1}}+\frac{2}{\sqrt{n^2+2}}+\cdots+\frac{n}{\sqrt{n^2+n}}\leq\frac{1}{\sqrt{n^2+1}}+\frac{2}{\sqrt{n^2+1}}+\cdots+\frac{n}{\sqrt{n^2+1}}$.
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