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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.

Explore the approximation ln(n!) ~ n ln n using Stirling's approximation. This solution provides a step-by-step breakdown of the mathematical concepts behind factorials, natural logarithms, and how Stirling's formula helps approximate large factorials. Suitable for undergraduate mathematics students.

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