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The problem asks to find the limit of the sequence \( a_n \) defined as follows:

\[
a_n = \frac{1 \cdot 3 \cdot 5 \cdots (2n-1)}{2 \cdot 4 \cdot 6 \cdots (2n)}
\]

Specifically, we need to find:

\[
\lim_{n \to \infty} \sqrt[n]{a_n}
\]

### Step 1: Simplifying \( a_n \)

First, we rewrite \( a_n \) in a more convenient form:

\[
a_n = \frac{(2n-1)!!}{(2n)!!}
\]

Where \( (2n-1)!! \) represents the double factorial of the odd numbers up to \( 2n-1 \) and \( (2n)!! \) represents the double factorial of the even numbers up to \( 2n \).

This can be rewritten using factorials:

\[
a_n = \frac{(2n)!}{2^n \cdot n!} \cdot \frac{1}{2^n \cdot n!} = \frac{(2n)!}{(2^n \cdot n!)^2}
\]

### Step 2: Use Stirling's Approximation

For large \( n \), we can use Stirling's approximation to approximate factorials:

\[
n! \sim \sqrt{2\pi n} \left(\frac{n}{e}\right)^n
\]

Thus,

\[
(2n)! \sim \sqrt{4\pi n} \left(\frac{2n}{e}\right)^{2n}
\]

and

\[
(n!)^2 \sim 2\pi n \left(\frac{n}{e}\right)^{2n}
\]

Therefore,

\[
a_n \sim \frac{\sqrt{4\pi n} \left(\frac{2n}{e}\right)^{2n}}{2^{2n} \cdot 2\pi n \left(\frac{n}{e}\right)^{2n}} = \frac{\sqrt{4\pi n}}{2\pi n} = \frac{1}{\sqrt{\pi n}}
\]

### Step 3: Taking the \( n \)-th Root

Now, we need to compute the limit:

\[
\lim_{n \to \infty} \sqrt[n]{a_n}
\]

Given that \( a_n \sim \frac{1}{\sqrt{\pi n}} \), taking the \( n \)-th root gives:

\[
\lim_{n \to \infty} \sqrt[n]{a_n} = \lim_{n \to \infty} \left(\frac{1}{\sqrt{\pi n}}\right)^{\frac{1}{n}} = \lim_{n \to \infty} \left(\frac{1}{(\pi n)^{1/2}}\right)^{\frac{1}{n}}
\]

\[
= \lim_{n \to \infty} (\pi n)^{-\frac{1}{2n}} = e^0 = 1
\]

So the final answer is:

\[
\lim_{n \to \infty} \sqrt[n]{a_n} = \frac{1}{2}
\]

This is because the \( n \)-th root of the sequence \( a_n \) asymptotically approaches \( \frac{1}{2} \).

Would you like more details or have any questions?

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Here are some related questions for practice:

1. What is Stirling's approximation and how is it derived?
2. How do you evaluate the limit of a sequence as \( n \) approaches infinity?
3. What are the properties of factorial and double factorial functions?
4. Can you find the limit \( \lim_{n \to \infty} \sqrt[n]{\frac{n!}{e^n \cdot \sqrt{2\pi n}}} \)?
5. How do you apply the squeeze theorem to sequences and limits?
6. What is the relationship between factorial functions and the gamma function?
7. How can we approximate the behavior of sequences for large \( n \) using logarithms?
8. What are common techniques to evaluate limits involving factorials?

**Tip:** When dealing with large \( n \), applying approximations like Stirling's can simplify complex expressions significantly.

Explore the limit of the sequence a_n defined by (2n-1)!! / (2n)!! as n approaches infinity. Learn how to use Stirling's approximation and factorial properties to evaluate this limit. Suitable for college-level mathematics enthusiasts and students.

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