SUMMARY
The equation \(\pi = 16\arctan\left(\frac{1}{5}\right) - 4\arctan\left(\frac{1}{239}\right)\) is derived from Machin's formula, which utilizes the series expansion of the arctangent function. The series expansion for \(\tan^{-1}(x)\) is \(\frac{\pi}{2} - \frac{1}{x} + \frac{1}{3x^3} - \frac{1}{5x^5} + \ldots\). This relationship is exact and can be verified through manipulation of the series. The discussion emphasizes the importance of understanding series expansions in trigonometric identities.
PREREQUISITES
- Understanding of Machin's formula
- Familiarity with series expansions in calculus
- Knowledge of arctangent function properties
- Basic skills in mathematical proof techniques
NEXT STEPS
- Study the derivation of Machin's formula in detail
- Explore the convergence of the arctangent series
- Learn about other formulas for calculating \(\pi\)
- Investigate the applications of arctangent in numerical methods
USEFUL FOR
Mathematicians, students of calculus, and anyone interested in the derivation of mathematical constants and their applications in numerical analysis.