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- Apr 13, 2013

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I have to check if the sequence $a_{n}=\frac{1}{\sqrt{n^2+1}}+\frac{2}{\sqrt{n^2+2}}+...+\frac{n}{\sqrt{n^2+n}}$ converges.I thought that:$$\frac{n^{2}(n+1)}{2\sqrt{n^2+n}} \leq a_{n} \leq \frac{n^{2}(n+1)}{2\sqrt{n^2+1}}$$ Because of the fact that:

$$\lim_{n \to \infty}\frac{n^{2}(n+1)}{2\sqrt{n^2+n}}=\lim_{n \to \infty}\frac{n^{2}(n+1)}{2\sqrt{n^2+1}}=\infty$$ I though that the sequence diverges.

Could you tell me if it is right?