Does this series converge? $\sum_{n=1}^{\infty} \left( \sqrt{n+\sqrt n} - \sqrt{n-\sqrt n} \right)$

Question

So I have following series:

$$\sum_{x=1}^{\infty}\left(\sqrt{x+\sqrt x}-\sqrt{x-\sqrt x}\right)$$

and I have to find out if it converges.

By Ratio Test it gave me 1, so no answer. Wolfram Alpha told me that it doesn't converge.
Does someone has an idea?

Greatings

Answer 1

Note: $$\sqrt{n + \sqrt{n}} - \sqrt{n - \sqrt{n}} = \frac{2\sqrt{n}}{\sqrt{n + \sqrt{n}} + \sqrt{n - \sqrt{n}}}$$ $$=\frac{2}{\sqrt{1+\frac{1}{\sqrt{n}}} + \sqrt{1-\frac{1}{\sqrt{n}}}}$$ Which will tend to $1$ as $n\to\infty$

Answer 2

Actually terms do not even tend to $0$. Since $x\mapsto\sqrt{x}$ is concave, we have $$ \begin{aligned} \sqrt{x+\sqrt{x}}-\sqrt{x-\sqrt{x}}&=\frac{2\sqrt{x}}{\sqrt{x+\sqrt{x}}+\sqrt{x-\sqrt{x}}}\\ &\geq\frac{\sqrt{x}}{\sqrt{x}}=1. \end{aligned} $$