SUMMARY
The closed form of the power series 1 + 3x + 6x² + 10x³ + 15x⁴ + 21x⁵ + ... can be derived by differentiating the geometric series formula. Starting with the series 1 + x + x² + ... = 1/(1-x), differentiating both sides yields the first derivative, which leads to the second derivative revealing a pattern. The coefficients of the series correspond to the triangular numbers, and the closed form is established as a function of x.
PREREQUISITES
- Understanding of power series and their convergence
- Familiarity with geometric series and their derivatives
- Knowledge of triangular numbers and their properties
- Basic calculus, specifically differentiation techniques
NEXT STEPS
- Study the derivation of closed forms for other power series
- Explore the properties of triangular numbers and their applications
- Learn about generating functions in combinatorics
- Investigate advanced techniques in series convergence and divergence
USEFUL FOR
Mathematics students, educators, and anyone interested in series convergence, combinatorial mathematics, or calculus applications.