Limit problem using a Riemann sum: $\lim_{n\to\infty}\frac{\sqrt{n+1}+\sqrt{n+2}+\dots+\sqrt{2n}}{\sqrt{1}+\sqrt{2}+\dots+\sqrt{n}}$

Question

I am trying to find

$$\lim_{n\to\infty}\frac{\sqrt{n+1}+\sqrt{n+2}+\dots+\sqrt{2n}}{\sqrt{1}+\sqrt{2}+\dots+\sqrt{n}}$$

I believe I should use that

$$\int_0^1 f(x)\,dx=\lim_{n\to\infty}\frac1n \sum_{k=1}^n f\left(\frac{k}n\right)$$

to find the limit, but I am lost. Can anyone help?

Answer 1

I believe I should use that

$$\int_0^1 f(x)\,dx=\lim_{n\to\infty}\frac1n \sum_{k=1}^n f\left(\frac{k}n\right)$$ to find the limit, but I am lost. Can anyone help?

Hint. One may write, as $n\to \infty$, $$ \begin{align} \frac{\sqrt{n+1}+\sqrt{n+2}+\dots+\sqrt{2n}}{\sqrt{1}+\sqrt{2}+\dots+\sqrt{n}}&=\frac{\frac1n\left(\sqrt{1+\frac1n}+\sqrt{1+\frac2n}+\dots+\sqrt{1+\frac nn}\right)}{\frac1n\left(\sqrt{\frac1n}+\sqrt{\frac2n}+\dots+\sqrt{\frac nn}\right)} \\\\&=\frac{\frac1n \sum_{k=1}^n \sqrt{1+\frac{k}n}}{\frac1n \sum_{k=1}^n \sqrt{\frac{k}n}}. \end{align} $$

Answer 2

If we define $a_{n}=\sum_{k=1}^{n}\sqrt{n+k}$ and $b_{n}=\sum_{k=1}^{n}\sqrt{k}$ we have $$\lim_{n\rightarrow\infty}\frac{a_{n+1}-a_{n}}{b_{n+1}-b_{n}}=\lim_{n\rightarrow\infty}\frac{\sqrt{2n+1}+\sqrt{2n+2}-\sqrt{n+1}}{\sqrt{n+1}}=2\sqrt{2}-1$$ hence, by Stolz-Cesàro theorem, $$\lim_{n\rightarrow\infty}\frac{a_{n}}{b_{n}}=\color{red}{2\sqrt{2}-1}.$$