SUMMARY
The Taylor series expansion for the function \( f(x) = x^4 - 3x^2 + 1 \) centered at \( a = 1 \) is given by \( -1 - 2(x-1) + 3(x-1)^2 + 4(x-1)^3 + (x-1)^4 \). Since this Taylor series is a polynomial, it converges for all values of \( x \). Therefore, the radius of convergence is \( (-\infty, \infty) \).
PREREQUISITES
- Understanding of Taylor series and polynomial functions
- Knowledge of power series expansions
- Familiarity with the concept of radius of convergence
- Basic calculus, particularly differentiation and evaluation of functions
NEXT STEPS
- Study the derivation of Taylor series for various functions
- Learn about the ratio test for determining the radius of convergence
- Explore the implications of polynomial convergence in real analysis
- Investigate applications of Taylor series in numerical methods
USEFUL FOR
Students and professionals in mathematics, particularly those studying calculus, real analysis, or numerical methods, will benefit from this discussion.