Equivalence of Definitions of Real Exponential Function/Power Series Expansion equivalent to Differential Equation

Sum of Series implies Solution of Differential Equation
Let $\exp x$ be the real function defined as the sum of the power series:


 * $\exp x := \displaystyle \sum_{n \mathop = 0}^\infty \frac {x^n} {n!}$

Let $y = \displaystyle \sum_{n \mathop = 0}^\infty \frac {x^n} {n!}$.

Then:

Setting $x = 0$ we find:

That is:

$\exp x$ is the solution of the differential equation:
 * $\dfrac {\d y} {\d x} = y$

satisfying the initial condition $f \left({0}\right) = 1$.

Solution of Differential Equation implies Sum of Series
Let $\exp x$ be the real function defined as the solution of the differential equation:
 * $\dfrac {\d y} {\d x} = y$

satisfying the initial condition $f \left({0}\right) = 1$.

We have Taylor Series Expansion for Exponential Function: