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Please suggest some hint to test the convergence of the following series $$\sum_{n=1}^\infty (-1)^n(\sqrt{n+1}-\sqrt n)$$

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    $\begingroup$ Do you have the Leibniz criterion at your disposal? $\endgroup$ – m_l Mar 11 '13 at 14:15
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Hint:

$$\sqrt{n+1}-\sqrt{n} = \frac{1}{\sqrt{n+1}+\sqrt{n}}$$

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  • $\begingroup$ Oh No ... then $\lim (\sqrt{n+1}-\sqrt{n}) = \lim\dfrac{1}{\sqrt{n+1}+\sqrt{n}}=0$ and $\dfrac{1}{\sqrt{n+1}+\sqrt{n}}\ge \dfrac{1}{\sqrt{n+2}+\sqrt{n+1}}.$ Thanks. $\endgroup$ – Sugata Adhya Mar 11 '13 at 14:21
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We have $$u_n=(-1)^n(\sqrt{n+1}-\sqrt{n})=\frac{(-1)^n}{\sqrt{n+1}+\sqrt{n}}$$ So the sequence $(|u_n|)_n$ converges to $0$ and is monotone decreasing then by Alternating series test the series $\sum_n u_n$ is convergent.

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