EDB — 1NJ

Exercises

1. [1NJ]Let \(I⊆ ℝ\) be a nonempty open interval. Let \(f:I→ ℝ\) be a \(C^∞\) class function. Let

   \[ b_ n=\sup _{x∈ I}|f^{(n)}(x)|=\| f^{(n)}\| _∞\quad ; \]

   if 

   \[ \limsup _{n→∞} \frac 1 n \sqrt[n]{b_ n} {\lt}∞ \]

   then \(f\) is analytic.

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

   Solution 1

   [1NK]↺↻