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?

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 $n$.