The equation shown in the image is the definition of the Euler-Mascheroni constant (denoted by $\gamma$):

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

Here is a breakdown:

• $\sum_{k=1}^{n} \frac{1}{k}$ is the harmonic series up to $n$, which is the sum of the reciprocals of the first $n$ positive integers.
• $\ln n$ is the natural logarithm of $n$.
• The Euler-Mascheroni constant, $\gamma$, is the limiting difference between the harmonic series and the natural logarithm as $n$ approaches infinity.

$\gamma$ is approximately equal to $0.57721$, but it has been studied extensively, and its exact value is unknown as it is believed to be an irrational number.

Would you like further details or have any questions?

Here are 5 questions that could further expand your understanding:

1. How is the Euler-Mascheroni constant related to integrals and special functions?
2. Can the harmonic series be expressed using an approximation for large $n$?
3. Why is $\gamma$ considered an important constant in number theory?
4. What are the applications of the Euler-Mascheroni constant in analysis?
5. How does the Euler-Mascheroni constant appear in probability and statistics?

Tip: The Euler-Mascheroni constant often appears in problems involving asymptotics, summation formulas, and integrals where logarithms and harmonic sums come into play.