SUMMARY
The integral $$\int^1_0 \left \{ \frac{1}{x} \right \} \, dx$$ evaluates to $$1 - \gamma$$, where $$\gamma$$ represents the Euler-Mascheroni constant. By applying the change of variable $$\frac{1}{x} = t$$, the integral transforms into $$\int_{1}^{\infty} \frac{\{t\}}{t^{2}} \, dt$$. This leads to the conclusion that the limit of the expression $$\lim_{n \rightarrow \infty} (\ln n - \sum_{k=2}^{n} \frac{1}{k})$$ results in the value $$1 - \gamma$$.
PREREQUISITES
- Understanding of integral calculus
- Familiarity with the concept of fractional parts
- Knowledge of the Euler-Mascheroni constant ($$\gamma$$)
- Experience with change of variables in integrals
NEXT STEPS
- Study the properties of the Euler-Mascheroni constant ($$\gamma$$)
- Learn about fractional part functions and their applications in integrals
- Explore advanced techniques in integral calculus, particularly change of variables
- Investigate the convergence of series and limits in calculus
USEFUL FOR
Mathematicians, students studying calculus, and anyone interested in advanced integral evaluation techniques.