Why is $\lim_{n \to +\infty }{\sqrt[n]{a_1 a_2 \cdots a_n}} =\lim_{n \to +\infty}{a_n}$

Question

Solving some problems regarding limits and sequence convergence, i stumbled upon a task, and it's solution relies on, and i quote: "We now use a well-known theorem : $$\lim_{n \to +\infty }{\sqrt[n]{a_1 a_2 \ldots a_n}} = \lim_{n \to +\infty}{a_n}$$

This isn't really intuitive (at least to me) and I don't know how to prove it. The original task was to find the limit of $$\lim_{n \to +\infty }{\sqrt[n]{\bigg{(}1+\frac{1}{1}\bigg{)} \bigg{(}1+\frac{1}{2}\bigg{)}^2 \ldots \bigg{(}1+\frac{1}{n}\bigg{)}^n}} $$ which of course, using the expression above is just $e$.

may I ask you which book you found this task in? I'm asking because it appeared among our exercise tasks and I still don't know how to prove this theorem (if it really is a theorem). It would really help me because I'm a first-year student and there are a very few books available in my mother-tongue. — PinkyWay, Dec 21 '19 at 08:33
See also: If $\lim_{n\to\infty}a_n=a$ and $a_n>0$ for all $n$, then we have $ \lim_{n\to\infty}\sqrt[n]{a_1a_2\cdots a_n}=a $? and If $(x_n) \to x$ then $(\sqrt[n]{x_1x_2\cdots x_n}) \to x$. — Martin Sleziak, Dec 22 '19 at 14:05
You need to assume more than just $a_n\gt0$ for all $n$. For example, if
$$a_n= \begin{cases} 1&\text{if $n$ is odd}\ 2&\text{if $n$ is even} \end{cases}$$

then $\lim_{n\to\infty}\sqrt[n]{a_1a_2\cdots a_n}=\sqrt2$ but $\lim_{n\to\infty}a_n$ does not exist. — Barry Cipra, May 28 '20 at 00:48

score 8 · Accepted Answer · answered Jan 25 '15 at 20:30

8

Take the logarithm of both sides. Then you want to prove

$$\lim_{n\rightarrow \infty} \frac{1}{n}\sum_1^n a_i = \lim_{n\rightarrow \infty} a_n.$$

This is a standard result about Cesàro means.

answered Jan 25 '15 at 20:30

Potato

40,171

score 3 · Answer 2 · answered Jan 25 '15 at 20:29

3

Take the log of the $n$-root, and applied the Cesaro theorem to it, showing that it will converge to the log of $(a_n)_n$'s limit. Take the exp to finish.

answered Jan 25 '15 at 20:29

Olórin

12,040

3

of course, continuity of log and exp is crucial – Olórin Jan 25 '15 at 20:30

Why is $\lim_{n \to +\infty }{\sqrt[n]{a_1 a_2 \cdots a_n}} =\lim_{n \to +\infty}{a_n}$

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