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

real-analysis sequences-and-series limits

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 $e$.


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.


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.

Related

Arrangement of Numbers
A binary operation, closed over the reals, that is associative, but not commutative
If $m$ and $n$ are distinct positive integers, then $m\mathbb{Z}$ is not ring-isomorphic to $n\mathbb{Z}$
Is $2 + 2 + 2 + 2 + ... = -\frac12$ or $-1$?
Show that $\int_{-\pi}^\pi\sin mx\sin nx d x$ is 0 $m\neq n$ and $\pi$ if $m=n$ using integration by parts
Derivative of $x^{x^{\cdot^{\cdot}}}$?
Is the complex Banach space $C([0,1])$ dual to any Banach Space?
Proving that a natural number made entirely of 6's and 0's is not a square.
Proof that $t-1-\log t \geq 0$ for $t > 0$
Finding real roots of $ P(x)=x^8 - x^7 +x^2 -x +15$
Standard machine in measure theory
Why are logarithms not defined for 0 and negatives?