PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Foundations of mathematical language · Foundations of mathematical language
Mathematical Methods for Physics, Ch Complex numbers, the complex exponential and phasors
Problem
The function f(x)=x2−2x+51 is finite and smooth for every real x, since x2−2x+5=(x−1)2+4 is never less than 4.
((a)) Find the radius of convergence of its series in powers of x, centred on x=0.
((b)) Find the radius of convergence of its series in powers of (x−1), centred on x=1, and confirm it with the geometric series of Chapter 2.
((c)) At x=2.5 the function has the ordinary value 0.16. Does each of the two series converge there? Explain why the answers differ although f is smooth everywhere on the real line.