What is the convergence or divergence of $\sum _{n=1}^{\infty \:}\left(\sqrt{n+\sqrt{n}}-\sqrt{n}\right)$

Question

$\sum _{n=1}^{\infty \:}\left(\sqrt{n+\sqrt{n}}-\sqrt{n}\right)$ Can you show me the work for this question

Answer 1

Multiply top and bottom by $\sqrt{n+\sqrt{n}}+\sqrt{n}$. You will see that the terms do not approach $0$, so the series diverges.

Answer 2

$$ \sqrt{n+\sqrt{n}} -\sqrt{n}= \sqrt{n}\left[\sqrt{1+\frac 1{\sqrt{n}}}-1\right] \\ \sim \sqrt{n}\frac 1{2\sqrt{n}} = \frac 12 $$ hence the series is divergent, the partial sum being $\simeq \frac n2$

Answer 3

You could look at the integral of:

$$ \int \sqrt{x + \sqrt{x}} dx $$

or even multiply 1, expressed as:

$$ \sqrt{n + \sqrt{n}} + \sqrt{n} $$

and look where this series goes.