EDB — 1M3

Exercises

1. [1M3]Prerequisites:[1K9]↺↻,[1KQ]↺↻, [20V]↺↻, [20W]↺↻.It is customary to define

   $$e^z={\sum }_{k=0}^{\infty }\frac{1}{k!}{z}^k$$

   for $z\in ℂ$. We want to reflect on this definition.

   • First, for each $z\in ℂ$, we can actually define

     $$f(z)={\sum }_{k=0}^{\infty }\frac{1}{k!}{z}^k$$

     (Note that the radius of convergence is infinite — as it easily occurs using the root criterion [219]↺↻).

   • We note that $f(0)=1$; we define $e=f(1)$ which is Euler’s number ¹

   • Show that $f(z+w)=f(z)f(w)$ for $z,w\in ℂ$.

   • It is easy to verify that $f(x)$ is monotonic increasing for $x\in (0,\infty )$; by the previous relation, $f(x)$ is monotonic increasing for $x\in ℝ$.

   • Then show that, for $n,m>0$ integer, $f(n/m)=e^{n/m}$ (for the definition of $e^{n/m}$ see [20V]↺↻).

   • Deduce that, for every $x\in ℝ$, $f(x)=e^x$ (for the definition of $e^x$ see [20W]↺↻)

   Solution 1

   [1M4]↺↻