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.