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.
2026-04-03 18:39:45.1775241585
Series convergence (or divergence)
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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 $n \to \infty$ .