The question is:

Prove that if $\lim\limits_{n\to\infty}X_n = a$, and $X_n > 0$, $n$ is any natural number, then $\lim\limits_{n\to\infty} \sqrt[n]{X_1\cdot X_2\cdot \cdot \cdot X_n} = a$.

I already get stuck on this question for half a day and can only get the following unsatisfactory result:

Consider $\sqrt[n]{X_1\cdot X_2\cdot \cdot \cdot X_n} = a \cdot \sqrt[n]{i_1\cdot i_2\cdot \cdot \cdot i_n}$ where ${i_n}$ converges to 1 as n tends to infinity.

Then I try to prove $\sqrt[n]{i_1\cdot i_2\cdot \cdot \cdot i_n}$ converges to 1 as well. However, utilizing the fact that ${i_n}$ converges to 1 only gets me to $\sqrt[n]{i_1\cdot i_2\cdot \cdot \cdot i_n} <\ (\epsilon + 1)(\epsilon + 1)^{\frac{n-N_0}{n}} \le \epsilon (\epsilon + 2) +1$.

Can anybody please help me understand how should I deal with the question? Thank you.

  • Taking logarithms is very often helpful when dealing with products. – Daniel Fischer Jan 31 '16 at 12:41
up vote 0 down vote accepted

Apply the theorem of Cesaro-Stolz to the logarithm of the sequence. This logarithm is the Cesaro mean $$ \frac{\log X_1+\log X_2+\dots+\log X_n}{n} $$

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