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I've got the following exercise:
Show that the series $\displaystyle\sum_{n=1}^{\infty}\dfrac{1}{\sqrt{n}+\sqrt{n+1}} $ converges and compute its limit if it's posible.
I've already tried with the Integral test but is worthless, I've got the same thought about using the ratio test. So I don't know which one of the convergence test would be most useful to compute the limit.

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  • $\begingroup$ Compare to $\sum_{n \geq 1} \frac{1}{\sqrt{n+1} + \sqrt{n+1}} = \infty$, by the integral test. $\endgroup$ – Dzoooks Jun 5 '18 at 2:11
  • $\begingroup$ $>\frac{1}{2\sqrt{n+1}}>\frac{1}{2(n+1)}$ $\endgroup$ – W. mu Jun 5 '18 at 2:11
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    $\begingroup$ I'd be shocked if it converges. $\endgroup$ – Randall Jun 5 '18 at 2:11
  • $\begingroup$ I know it's not the problem, but an alternative interesting question would be to replace the $+$ between the radicals with a $-$. That one also has an answer... $\endgroup$ – imranfat Jun 5 '18 at 2:15
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hint: Use comparison test with the series $\displaystyle \sum_{n=1}^\infty \dfrac{1}{\sqrt{n}}$, and this one is for sure divergent. Or use the definition of the partial sum $S_n = \displaystyle \sum_{k=1}^n \dfrac{1}{\sqrt{k}+\sqrt{k+1}} = \displaystyle \sum_{k=1}^n \left(\sqrt{k+1} - \sqrt{k}\right) = \sqrt{n+1} - 1\to \infty$ for $n \to \infty$ .

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  • $\begingroup$ Hello my friend. It's good to see you're back. (+1) I would have used your second approach (i.e., Rationalizing the denominator and evaluating the telescoping series). $\endgroup$ – Mark Viola Jun 5 '18 at 2:24
  • $\begingroup$ @MarkViola: Hi Mark, nice to see you back as well. Does Houston completely rebuild after Harvey? Is life back to normal ? $\endgroup$ – DeepSea Jun 5 '18 at 2:34
  • $\begingroup$ I live in a master planned community called The Woodlands, which is 45 miles North of downtown Houston. The Woodlands was largely unscathed. But in other areas of the Metropolis, there are areas that are still rebuilding. But for the most part, life is back to normal. $\endgroup$ – Mark Viola Jun 5 '18 at 2:40
  • $\begingroup$ I have the thought of moving back to Houson ( I used to live in Houston back in 2000 ) and get a small property there with KB Home. $\endgroup$ – DeepSea Jun 5 '18 at 2:42
  • $\begingroup$ It's a good place to live if you don't mind the hot, humid summers. $\endgroup$ – Mark Viola Jun 5 '18 at 2:43

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