Evaluate $$\lim_{n\to\infty}\frac{\sum_{k=1}^n\sqrt{k}}{\sum_{k=1}^n\sqrt{n+k}}$$
I can't seem to remove the $\frac{1}{n}$ term in order to use the riemann integral any help?
2026-04-04 13:37:28.1775309848
Using Riemann integrals to find limits
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Hint:
Let me introduce some terms $$\lim_{n\to\infty}\frac{\sum_{k=1}^n\sqrt{k}}{\sum_{k=1}^n\sqrt{n+k}} = \lim_{n \to \infty}\frac{\frac1n\sum_{k=1}^n\sqrt{\frac{k}{n}}}{\frac1n\sum_{k=1}^n\sqrt{\frac{n+k}{n}}} $$