The expression shown in the image seems to be asking for an approximation of $\ln(n!)$, which is the natural logarithm of $n!$, where $n!$ (n factorial) is the product of all positive integers up to $n$. The question is whether this is approximately equal to $n \ln n$.

To answer this, we can apply Stirling's approximation, which is commonly used to approximate the factorial function for large $n$. Stirling's approximation states:

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

Taking the natural logarithm of both sides gives:

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

We can break this down further:

$\ln(n!) \sim \ln(\sqrt{2 \pi n}) + \ln\left(\frac{n}{e}\right)^n$

The first term becomes:

$\ln(\sqrt{2 \pi n}) = \frac{1}{2} \ln(2 \pi n)$

The second term simplifies to:

$\ln\left(\frac{n}{e}\right)^n = n \ln\left(\frac{n}{e}\right) = n \ln n - n$

Thus, the overall approximation is:

$\ln(n!) \sim \frac{1}{2} \ln(2 \pi n) + n \ln n - n$

For large $n$, the term $\frac{1}{2} \ln(2 \pi n)$ becomes negligible in comparison to $n \ln n$ and $n$, so the leading approximation becomes:

$\ln(n!) \sim n \ln n - n$

This shows that $\ln(n!)$ is approximately $n \ln n$, but it also includes an additional $-n$ term. Therefore, $\ln(n!)$ is not exactly $n \ln n$, but for large values of $n$, it is asymptotically close to $n \ln n$.

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Do you need further clarifications or details on this solution?

Here are five related questions for further exploration:

1. How does Stirling’s approximation improve as $n$ increases?
2. What is the significance of the $-n$ term in the Stirling approximation?
3. How can Stirling’s approximation be used to estimate large factorials?
4. Why does the $\frac{1}{2} \ln(2 \pi n)$ term become negligible for large $n$?
5. How does the approximation $\ln(n!) \sim n \ln n - n$ compare with other approximations for factorials?

Tip: Stirling's approximation is particularly useful in combinatorics and statistical mechanics, where the factorial of large numbers frequently appears.