The expression provided in the image represents Euler's constant $\gamma$, also known as the Euler-Mascheroni constant. It is defined as the 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 n-th harmonic number.
• $\ln n$ is the natural logarithm of $n$.
• The limit expresses the difference between the harmonic number and the natural logarithm of $n$, as $n$ approaches infinity. This difference converges to a constant, $\gamma$.

Known Approximation of $\gamma$:

$\gamma \approx 0.57721$

Euler's constant frequently appears in number theory and analysis, particularly in integrals, sums, and asymptotic formulas.

Would you like further details or have any questions?

Here are 5 related questions to expand your understanding:

1. How can the Euler-Mascheroni constant be derived from integrals?
2. What is the significance of the harmonic series in this formula?
3. Can you provide the relationship between $\gamma$ and the digamma function?
4. How does Euler's constant appear in the Riemann zeta function?
5. What are some numerical methods for approximating Euler's constant?

Tip: The Euler-Mascheroni constant plays a crucial role in advanced calculus, particularly in series and integral convergence analyses.