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

## Theorem

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

### As a Power Series Expansion

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

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

### As the Solution of a Differential Equation

The **exponential function** can be defined as the unique solution $y = \map f x$ to the first order ODE:

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

satisfying the initial condition $\map f 0 = 1$.

## Proof

#### Power Series Expansion implies Solution of Differential Equation

Let $\exp x$ be the real function defined as the sum of the power series:

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

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

Then:

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

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

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

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

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

\(\ds \) | \(=\) | \(\ds y\) |

Setting $x = 0$ we find:

\(\ds \map y 0\) | \(=\) | \(\ds \sum_{n \mathop = 0}^\infty \frac {0^n} {n!}\) | ||||||||||||

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

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

\(\ds \) | \(=\) | \(\ds 1\) | Definition of $0^0$ |

$\blacksquare$

That is:

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

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

satisfying the initial condition $\map f 0 = 1$.

$\Box$

#### Solution of Differential Equation implies Power Series Expansion

Let $\exp x$ be the real function defined as the particular solution of the differential equation:

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

satisfying the initial condition $\map f 0 = 1$.

We have Taylor Series Expansion for Exponential Function:

From Higher Derivatives of Exponential Function, we have:

- $\forall n \in \N: \map {f^{\paren n} } {\exp x} = \exp x$

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

- $\displaystyle \exp x = 1 + \frac x {1!} + \frac {x^2} {2!} + \cdots + \frac {x^{n - 1} } {\paren {n - 1}!} + \frac {x^n} {n!} \map \exp \eta$

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

Hence:

\(\ds \size {\exp x - \paren {1 + \frac x {1!} + \frac {x^2} {2!} + \cdots + \frac {x^{n - 1} } {\paren {n - 1}!} } }\) | \(=\) | \(\ds \size {\frac {x^n} {n!} \map \exp \eta}\) | ||||||||||||

\(\ds \) | \(\le\) | \(\ds \frac {\size {x^n} } {n!} \map \exp {\size x}\) | Exponential is Strictly Increasing | |||||||||||

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

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

The result follows.

$\blacksquare$