EDB — 1NG

Exercises

1. [1NG] Let \(f:ℝ→ℝ\) be a \(C^∞\) class function; fix \(x_ 0∈ ℝ\) and define

   \[ g(x)= ∑_{n=0}^∞\frac{f^{(n)}(x_ 0)}{n!} (x-x_ 0)^ n \]

   using the Taylor series; suppose \(g\) has radius of convergence \(R{\gt}0\): So \(g:J→ℝ\) is a well-defined function, where \(J=(x_ 0-R,x_ 0+R)\). Can it happen that \(f(x)≠ g(x)\) for a point \(x∈ J\)?

   And if \(f\) is analytic? ¹

   Solution 1

   [1NH]↺↻