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The question in the image seems to be asking whether the factorial \( n! \) is asymptotically equivalent to the expression \( \left( \frac{n+1}{2} \right)^n \). The symbol \( \sim \) indicates an asymptotic approximation, meaning we are interested in whether \( n! \) behaves similarly to \( \left( \frac{n+1}{2} \right)^n \) for large values of \( n \).

### Comparing \( n! \) with \( \left( \frac{n+1}{2} \right)^n \):

To analyze this, let's compare these two expressions asymptotically.

#### Stirling's Approximation:
A well-known approximation for large \( n \) is Stirling's approximation for factorials:
\[
n! \sim \sqrt{2\pi n} \left( \frac{n}{e} \right)^n
\]
This provides a good approximation of the growth rate of \( n! \).

#### Expression \( \left( \frac{n+1}{2} \right)^n \):
The expression in question is:
\[
\left( \frac{n+1}{2} \right)^n
\]

For large \( n \), this behaves similarly to:
\[
\left( \frac{n}{2} \right)^n
\]

### Comparing Growth Rates:
- \( n! \sim \sqrt{2\pi n} \left( \frac{n}{e} \right)^n \)
- \( \left( \frac{n+1}{2} \right)^n \sim \left( \frac{n}{2} \right)^n \)

We see that the growth rates are not exactly the same because Stirling's approximation has an additional factor of \( \frac{n}{e} \) in the exponent, while the expression \( \left( \frac{n+1}{2} \right)^n \) grows more slowly due to the factor of \( \frac{n+1}{2} \) instead of \( \frac{n}{e} \).

### Conclusion:
No, the approximation \( n! \sim \left( \frac{n+1}{2} \right)^n \) is not true. The factorial \( n! \) grows faster than the expression \( \left( \frac{n+1}{2} \right)^n \) because of the difference in the factors inside the exponent.

Would you like further details or clarification on any part of this?

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Here are 5 related questions that expand on this:

1. What is Stirling's approximation, and how is it derived?
2. How does the growth of factorials compare with exponential functions?
3. Can the approximation \( n! \sim \left( \frac{n+1}{2} \right)^n \) be useful in any specific cases?
4. How do other asymptotic approximations compare with Stirling's approximation?
5. What are the applications of asymptotic approximations like this in real-world problems?

**Tip:** When comparing two functions asymptotically, always examine the dominant term in their growth rates for large values of \( n \).

Is n! asymptotically equivalent to (n+1)/2 raised to the power n?

Explore whether the factorial n! is asymptotically equivalent to (n+1)/2 raised to the power n using Stirling's approximation. This analysis compares the growth rates of the factorial and exponential terms, highlighting key mathematical concepts and formulas.

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