the limit of $\frac{1}{\sqrt{n}}(1+\frac{2}{1+\sqrt2}+\frac{3}{1+\sqrt2+\sqrt3}+\dots+\frac{n}{1+\sqrt2+\sqrt3+\dots+\sqrt n})$ as $n\to\infty$

The Stolz–Cesàro theorem indeed suffices, with a derivative-like limit (the last one below).

Let $a_n=\sum\limits_{k=1}^n\sqrt{k}$ and $b_n=\sum\limits_{k=1}^n k/a_k$; we're computing \begin{align*} \lim_{n\to\infty}\frac{b_n}{\sqrt{n}} &=\lim_{n\to\infty}\frac{b_n-b_{n-1}}{\sqrt{n}-\sqrt{n-1}} \\&=\lim_{n\to\infty}\frac{n}{a_n}(\sqrt{n}+\sqrt{n-1}) \\&=2\lim_{n\to\infty}\frac{n^{3/2}}{a_n} \\&=2\lim_{n\to\infty}\frac{n^{3/2}-(n-1)^{3/2}}{a_n-a_{n-1}} \\&=2\lim_{n\to\infty}\frac{n^{3/2}-(n-1)^{3/2}}{\sqrt{n}} \\&=\color{gray}{2\lim_{n\to\infty}\frac{1-(1-1/n)^{3/2}}{1/n}}=\mathbf{3}. \end{align*}