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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1Hint: What is the limit of the sequence $(\pi)^{1/n}$? – Nov 12 '13 at 06:39
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2Hint: $\pi$ is a red herring. – Nov 12 '13 at 06:41
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1For $k\gt 1000$, $\sqrt[k]{\pi}$ 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 06:52
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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 07:04
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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 07:05
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The limit should be one. – Mhenni Benghorbal Nov 12 '13 at 07:08
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I see. But how can I get the upper limit if I want to use squeeze theorem? I tried a lot of methods but failed. – Jason Nov 12 '13 at 07:28
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You should prove the more general statement: if $a_n$ is any sequence that converges to the finite number $L$, then $\lim_{n\to\infty} \frac{a_1+a_2+\cdots+a_n}n = L$. – Greg Martin Nov 12 '13 at 07:46
