# Equivalence of Definitions of Real Exponential Function/Sum of Series equivalent to Differential Equation

## Contents

## Theorem

The following definitions of the concept of **real exponential function** are equivalent:

### As the Sum of a Series

The **exponential function** can be defined as a power series:

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

### As the Solution of a Differential Equation

The **exponential function** can be defined as the unique solution $y = f \left({x}\right)$ to the first order ODE:

- $\dfrac {\d y} {\d x} = y$

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

## Proof

#### 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:

\(\displaystyle \dfrac {\d y} {\d x}\) | \(=\) | \(\displaystyle \dfrac \d {\d x} \sum_{n \mathop = 0}^\infty \frac {x^n} {n!}\) | $\quad$ | $\quad$ | |||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \dfrac \d {\d x} \paren {\frac {x^0} {0!} + \sum_{n \mathop = 1}^\infty \frac {x^n} {n!} }\) | $\quad$ extracting the zeroth term | $\quad$ | |||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle 0 + \sum_{n \mathop = 1}^\infty \frac {n x^{n - 1} } {n!}\) | $\quad$ Power Rule for Derivatives and Derivative of Constant | $\quad$ | |||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \sum_{n \mathop = 1}^\infty \frac {x^{n - 1} } {\paren {n - 1}!}\) | $\quad$ simplifying | $\quad$ | |||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \sum_{n \mathop = 0}^\infty \frac {x^n} {n!}\) | $\quad$ Translation of Index Variable of Summation | $\quad$ | |||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle y\) | $\quad$ | $\quad$ |

Setting $x = 0$ we find:

\(\displaystyle y \paren 0\) | \(=\) | \(\displaystyle \sum_{n \mathop = 0}^\infty \frac {0^n} {n!}\) | $\quad$ | $\quad$ | |||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \frac {0^0} {0!} + \sum_{n \mathop = 1}^\infty \frac {0^n} {n!}\) | $\quad$ | $\quad$ | |||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \frac {0^0} {0!}\) | $\quad$ as $0^n = 0$ for all $n > 0$ | $\quad$ | |||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle 1\) | $\quad$ Definition of $0^0$ | $\quad$ |

$\blacksquare$

That is:

$\exp x$ is the solution of the differential equation:

- $\dfrac {\d y} {\d x} = y$

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

$\Box$

#### 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:

From Higher Derivatives of Exponential Function, we have:

- $\forall n \in \N: f^{\left({n}\right)} \left({\exp x}\right) = \exp x$

Since $\exp 0 = 1$, the Taylor series expansion for $\exp x$ about $0$ is given by:

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

From Radius of Convergence of Power Series over Factorial, we know that this power series expansion converges for all $x \in \R$.

From Taylor's Theorem, we know that

- $\displaystyle \exp x = 1 + \frac x {1!} + \frac {x^2} {2!} + \cdots + \frac {x^{n - 1}} {\left({n - 1}\right)!} + \frac {x^n} {n!} \exp \left({\eta}\right)$

where $0 \le \eta \le x$.

Hence:

\(\displaystyle \left \vert {\exp x - \left({1 + \frac x {1!} + \frac {x^2} {2!} + \cdots + \frac {x^{n - 1} } {\left({n - 1}\right)!} }\right)} \right \vert\) | \(=\) | \(\displaystyle \left \vert {\frac {x^n} {n!} \exp \left({\eta}\right)} \right \vert\) | $\quad$ | $\quad$ | |||||||||

\(\displaystyle \) | \(\le\) | \(\displaystyle \frac {\left \vert {x^n} \right \vert} {n!} \exp \left({\left \vert {x} \right \vert}\right)\) | $\quad$ Exponential is Strictly Increasing | $\quad$ | |||||||||

\(\displaystyle \) | \(\to\) | \(\displaystyle 0 \text { as } n \to \infty\) | $\quad$ Series of Power over Factorial Converges | $\quad$ |

So the partial sums of the power series converge to $\exp x$.

The result follows.

$\blacksquare$