Exercises
[1M3]Prerequisites:[1K9],[1KQ], [20V], [20W].It is customary to define
\[ e^ z =∑_{k=0}^∞ \frac 1{k!} z^ k \]for \(z∈ℂ\). We want to reflect on this definition.
First, for each \(z∈ℂ\), we can actually define
\[ f(z) =∑_{k=0}^∞ \frac 1{k!} z^ k \](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 \(z,w∈ℂ\).
It is easy to verify that \(f(x)\) is monotonic increasing for \(x∈(0,∞)\); by the previous relation, \(f(x)\) is monotonic increasing for \(x∈ℝ\).
Then show that, for \(n,m{\gt}0\) integer, \(f(n/m) = e^{n/m}\) (for the definition of \(e^{n/m}\) see [20V]).
Deduce that, for every \(x∈ℝ\), \(f(x) = e^{x}\) (for the definition of \(e^{x}\) see [20W])
Solution 1