# Find the limit (if it exist) $\lim_{n \rightarrow \infty}\frac{\pi+\sqrt{\pi}+\cdots+\sqrt[n]{\pi}}{n}$

Find the limit (if it exist) $\lim_{n \rightarrow \infty}\frac{\pi+\sqrt{\pi}+\cdots+\sqrt[n]{\pi}}{n}$

• Hint: What is the limit of the sequence $(\pi)^{1/n}$? – user99914 Nov 12 '13 at 6:39
• Hint: $\pi$ is a red herring. – Hurkyl Nov 12 '13 at 6:41
• For $k\gt 1000$, $\sqrt[k]{\pi}$ is close to $1$. For $n\gt 1,000,000$, the part from $k=1$ to $k=1000$ makes no big difference. – André Nicolas Nov 12 '13 at 6:52
Following Greg's idea, note that $\lim\limits_{n\to\infty}\pi^{1/n}=1$ and use the result of this post.