Does the following series converge or diverge?
$$
\sum\limits_{n=1}^\infty\frac{1}{\sqrt{n}+\sqrt{n+1}}
$$
The methods I have at my disposal are geometric and harmonic series, comparison test, limit comparison test, and the ratio test.
Answer
It is not hard to see that
$$\sum_{n=1}^\infty\frac{1}{\sqrt{n+1}+\sqrt{n}}=\sum_{n=1}^\infty(\sqrt{n+1}-\sqrt{n})$$
As you know this series is divergent.
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