https://file.aiquickdraw.com/mathbot/file/172703803021033xbugbn.jpg

The expression in the image represents the following infinite product limit:

\[
\lim_{n \to \infty} \prod_{k=2}^{n} \left(1 - \frac{1}{k^2}\right)
\]

The general term in this product is \( \left( 1 - \frac{1}{k^2} \right) \), where \( k \) ranges from 2 to \( n \), and the product continues as \( n \) approaches infinity.

### Simplification:
Each term in the product can be simplified as:

\[
1 - \frac{1}{k^2} = \frac{k^2 - 1}{k^2} = \frac{(k-1)(k+1)}{k^2}
\]

Thus, the entire product can be written as:

\[
\prod_{k=2}^{n} \frac{(k-1)(k+1)}{k^2}
\]

This product can be simplified further by recognizing it as a telescoping product:

\[
\frac{1}{2} \cdot \frac{3}{2} \cdot \frac{4}{3} \cdot \frac{5}{4} \cdot \ldots \cdot \frac{n}{n-1} \cdot \frac{n+1}{n}
\]

The terms cancel out, leaving only the first few and last terms:

\[
\frac{1}{2} \cdot \frac{n+1}{n}
\]

### Taking the Limit:
As \( n \to \infty \), the last term \( \frac{n+1}{n} \) approaches 1. Therefore, the limit of the product is:

\[
\lim_{n \to \infty} \frac{1}{2} \cdot \frac{n+1}{n} = \frac{1}{2}
\]

Thus, the value of the infinite product is:

\[
\boxed{\frac{1}{2}}
\]

Would you like further details or have any questions?

### Related Questions:
1. How does the telescoping effect work in infinite product problems?
2. What are some other examples of telescoping products in calculus?
3. Can we apply this technique to other similar infinite product limits?
4. How is this related to series summation techniques?
5. What happens if the exponent in the denominator changes to something other than 2?

### Tip:
Telescoping products often simplify when successive terms cancel out, so identifying such patterns can significantly reduce the complexity of problems.

What is the limit of the infinite product \( \lim_{n \to \infty} \prod_{k=2}^{n} \left(1 - \frac{1}{k^2}\right) \)?

Explore the infinite product limit \( \lim_{n \to \infty} \prod_{k=2}^{n} \left(1 - \frac{1}{k^2}\right) \), learn how it simplifies via telescoping product techniques, and understand why the final value is \( \frac{1}{2} \). This detailed explanation covers key calculus concepts.

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