https://file.aiquickdraw.com/mathbot/file/1726942207229lxklrcg1.jpg

The expression in the image represents the **Euler-Mascheroni constant** \( \gamma \), defined by the following limit:

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

### Explanation:
- \( \sum_{k=1}^{n} \frac{1}{k} \) is the **harmonic series** up to \( n \), denoted as \( H_n \).
- \( \ln n \) is the natural logarithm of \( n \).
- The difference between the harmonic series and the natural logarithm approaches a constant as \( n \to \infty \), which is the Euler-Mascheroni constant \( \gamma \approx 0.57721 \).

The Euler-Mascheroni constant arises in various fields such as number theory, calculus, and analysis, particularly in the study of harmonic series and logarithmic integrals.

Would you like to delve into more details or have any specific questions?

Here are 5 related questions to explore:
1. What is the relationship between the harmonic series and the logarithm?
2. Why does the harmonic series grow without bound while the difference with \( \ln n \) converges?
3. How is \( \gamma \) connected to the zeta function and prime number theory?
4. What are some applications of the Euler-Mascheroni constant in real analysis?
5. How can we approximate \( \gamma \) using numerical methods?

**Tip:** Understanding how series behave as \( n \) approaches infinity is crucial in analyzing limits and sums in calculus.

What is the definition of the Euler-Mascheroni constant based on the limit involving the harmonic series and the natural logarithm?

Explore the definition of the Euler-Mascheroni constant through the limit of the harmonic series minus the natural logarithm as n approaches infinity. This problem introduces key concepts in calculus and number theory.

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