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The image you've provided shows the mathematical expression for the **Euler-Mascheroni constant** \( \gamma \), which is defined as the limiting difference between the harmonic series and the natural logarithm. The formula for \( \gamma \) is:

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

### Explanation:
- The **harmonic series** \( \sum_{k=1}^{n} \frac{1}{k} \) represents the sum of the reciprocals of the first \( n \) natural numbers.
- \( \ln n \) is the natural logarithm of \( n \).
- As \( n \) approaches infinity, the harmonic series grows without bound, but the difference between the harmonic series and the logarithm of \( n \) approaches a constant value, which is denoted by \( \gamma \).

This constant \( \gamma \) has an approximate value of \( \gamma \approx 0.57721 \).

### Questions for further exploration:
1. How is the Euler-Mascheroni constant used in number theory?
2. Can we derive an alternative expression for \( \gamma \) using integrals?
3. How does the harmonic series behave as \( n \to \infty \)?
4. Is the Euler-Mascheroni constant a rational or irrational number? 
5. What are some applications of \( \gamma \) in mathematical analysis and calculus?

**Tip:** The Euler-Mascheroni constant appears in many fields of mathematics, including asymptotic analysis, special functions, and number theory, particularly in the study of prime numbers.

What is the Euler-Mascheroni constant, and how is it defined mathematically?

Understand the definition of the Euler-Mascheroni constant, γ, as the limiting difference between the harmonic series and the natural logarithm. Explore its significance in mathematics, particularly in number theory and calculus, with a detailed formula and explanation.

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