1

Suppose that a sequence $a_n$ of positive numbers converges to $a$. Show that $$\lim_{n\rightarrow \infty}\left(\prod_{i=1}^{n}a_i\right)^{1/n}=a$$ This seems to be simple using that $x=e^{\log x}$, but I can't go any further after using that.

PS. I'm almost sure that this post is a duplicate, but I wasn't able to find it here. I'm sorry if that's the case.

user119459
  • 302
  • 1
  • 11
  • https://math.stackexchange.com/questions/2441672/show-that-lim-n-to-infty-sqrt-sum-limitsk-0n-lambda-k-prod-limits – Guy Fsone Jan 16 '18 at 20:10
  • https://math.stackexchange.com/questions/2440333/general-cesaro-lim-limits-n-to-infty-frac1-sum-limitsk-0n-lambda-k?rq=1 – Guy Fsone Jan 16 '18 at 20:11

1 Answers1

3

Hint 1: $$\ln \left(\prod_{i=1}^{n}a_i\right)^{1/n} = \frac{\ln(a_1)+\ln(a_2)+...+\ln(a_n)}{n}$$

Hint 2: Suppose that $\lim_n b_n =l$ where $l$ is real number or $\pm \infty$. Prove that $$ \lim_n \frac{b_1+b_2+...+b_n}{n}=l $$

This is a particular case of STolz-Cezaro, but easier to prove than the general version.

N. S.
  • 132,525
  • Thank you very much. I wasn't aware of the second hint/result, but now I can prove it and finish the first proof. – user119459 Apr 04 '15 at 18:15
  • @user119459 see also https://math.stackexchange.com/questions/2440333/general-cesaro-lim-limits-n-to-infty-frac1-sum-limitsk-0n-lambda-k?rq=1 – Guy Fsone Jan 16 '18 at 20:11