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

I have no idea about this.

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Hint: What is the limit of the sequence $(\pi)^{1/n}$? –  John 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
You could replace Pi by any number, the limit will always be 1 (1+ if the number is greater than 1, 1- if the number is lower than 1) –  Claude Leibovici Nov 12 '13 at 7:04
I can find the upper limit \pi and the lower limit 1, but I have no idea the next step. –  Jason Nov 12 '13 at 7:05

1 Answer 1

Following Greg's idea, note that $\lim\limits_{n\to\infty}\pi^{1/n}=1$ and use the result of this post.

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