How to investigate the convegence of this sequence?

Question

How to investigate the convegence of the following sequence using Cauchy's convergence test?

$$x_{n} = \frac{1}{1}+ \frac{1}{\sqrt[1]{2}+\sqrt{1}}+\frac{1}{\sqrt[2]{3}+\sqrt{2}}+ ... +\frac{1}{\sqrt[n-1]{n}+\sqrt{n-1}}$$

Answer 1

$(x_n)$ does not converge. To see this, notice that $$ \begin{split} x_n &> \frac11+\frac1{\sqrt2+1}+\frac1{\sqrt3+\sqrt2}+\cdots\\ &>\frac12+\frac1{2\sqrt2}+\frac1{2\sqrt3}\\ &=\frac12\sum_1^\infty\frac1{\sqrt n}. \end{split}$$ For any positive integer $n\geq\sqrt n$, so $\frac1n\leq\frac1{\sqrt n}$. Thus $$ x_n>\frac12\sum_1^\infty\frac1{\sqrt n}>\frac12\sum_1^\infty\frac1n. $$ The rightmost sum is (half of) the harmonic series, which is well known not to converge. Since $x_n$ is always larger than it, then surely it does not converge as well.