EDB — 1NJ

Exercises

1. [1NJ]Let $I\subseteq ℝ$ be a nonempty open interval. Let $f:I\to ℝ$ be a $C^{\infty }$ class function. Let

   $$b_n=\underset{x\in I}{sup}|f^{(n)}(x)|=\Vert f^{(n)}{\Vert }_{\infty };$$

   if 

   $$\underset{n\to \infty }{lim sup}\frac{1}{n}\sqrt[n]{b_n}<\infty$$

   then $f$ is analytic.

   Show with a simple example that the request is not necessary.

   Solution 1

   [1NK]↺↻