https://file.aiquickdraw.com/mathbot/file/1729622381089q30eki0x.jpg

The expression shown in the image represents the definition of the **Euler-Mascheroni constant**, denoted by \( \gamma \), which is given as:

\[
\gamma = \lim_{n \to \infty} \left( \sum_{k=1}^{n} \frac{1}{k} - \ln n \right)
\]

### Step-by-Step Explanation:

1. **Harmonic Series**: The term \( \sum_{k=1}^{n} \frac{1}{k} \) is the partial sum of the harmonic series. As \( n \to \infty \), this sum grows without bound, but it does so slowly.

2. **Natural Logarithm Approximation**: The logarithmic term \( \ln n \) serves as an approximation to the harmonic series. As \( n \to \infty \), the sum \( \sum_{k=1}^{n} \frac{1}{k} \) behaves similarly to \( \ln n \), but the difference between these two grows very slowly and tends towards a constant.

3. **Euler-Mascheroni Constant**: This limit describes the constant difference between the harmonic series and the natural logarithm as \( n \) approaches infinity. The value of this constant is approximately:
   
   \[
   \gamma \approx 0.57721
   \]

### Conclusion:

The constant \( \gamma \) is known as the **Euler-Mascheroni constant**, and its exact value is defined by the limit:

\[
\gamma = \lim_{n \to \infty} \left( \sum_{k=1}^{n} \frac{1}{k} - \ln n \right) \approx 0.57721
\]

Would you like further details or any additional clarifications?

Here are five related questions for further exploration:

1. How is the Euler-Mascheroni constant applied in number theory?
2. What is the difference between the harmonic series and \( \ln n \)?
3. Can the Euler-Mascheroni constant be represented as a continued fraction?
4. How does the Euler-Mascheroni constant appear in integrals and asymptotic expansions?
5. What is the relationship between the Euler-Mascheroni constant and the Riemann zeta function?

**Tip**: The Euler-Mascheroni constant is an irrational number, meaning it cannot be expressed as a simple fraction.

The expression represents the definition of the Euler-Mascheroni constant.

Explore the definition of the Euler-Mascheroni constant with step-by-step analysis. This problem involves limits, harmonic series, and logarithmic approximations. Suitable for advanced calculus students.

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