1
$\begingroup$

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.

$\endgroup$
  • 1
    $\begingroup$ Hint: What is the limit of the sequence $(\pi)^{1/n}$? $\endgroup$ – user99914 Nov 12 '13 at 6:39
  • 2
    $\begingroup$ Hint: $\pi$ is a red herring. $\endgroup$ – Hurkyl Nov 12 '13 at 6:41
  • 1
    $\begingroup$ 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. $\endgroup$ – André Nicolas Nov 12 '13 at 6:52
  • $\begingroup$ 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) $\endgroup$ – Claude Leibovici Nov 12 '13 at 7:04
  • $\begingroup$ I can find the upper limit \pi and the lower limit 1, but I have no idea the next step. $\endgroup$ – Jason Nov 12 '13 at 7:05
1
$\begingroup$

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

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.